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Magnetic and structural phase transitions in a dilute magnet with configurational degrees of freedom
Physica A 166 (1990) 157-171 North-Holland
MAGNETIC AND STRUCTURAL PHASE TRANSITIONS IN A DILUTE M A G N E T W I T H CONFIGURATIONAL DEGREES OF F R E E D O M I. THE GROUND STATE
H. JARZI~CKA and B. F E C H N E R Institute of Physics, Adam Mickiewicz University, Poznati, Poland
Received 22 November 1989 An annealed binary system with one magnetic component is considered. Chemical and magnetic interactions are taken into account which for some signs of their parameters induce two kinds of competitions between the magnetic atoms. The ground state versus the concentration of magnetic atoms in the mean field approximation is discussed. Characteristic diagrams are presented exposing the connection between magnetic and structural orders.
I. I n t r o d u c t i o n
The interrelation between magnetism and spatial distribution of atoms in a lattice has been investigated in many alloys and compounds. The best known compounds showing structural order are those of the types F e - C o [1, 2], F e - V [3, 4], F e - A I [5] crystallizing in the BCC lattice and F e - N i , F e - P d , F e - P t , M n - N i [6] crystallizing in the FCC lattice. On the other hand, a tendency to phase separation was observed in the alloys A u - F e [7], F e - C r [8], C u - N i , A u - N i , C u - P t [9]. A n o t h e r group of compounds including such alloys as Fe-A1, F e - N i may be structurally ordered or separated depending on the temperature and concentration of one of their components. Rich experimental evidence resulted in many theoretical works, which may be divided into three groups: those devoted to the studies of structural order, to phase separation and those studying the possibilities of coexistence of structural order and phase separation. At first structural order has been studied in dilute binary alloys with one magnetic component in the mean field approximation [5, 10, 11], in the Bethe approximation [12] and using the G r e e n function m e t h o d [13]. Then it has been studied in binary systems of two magnetic components in the mean field 0378-4371/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
158
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
approximation [1, 2, 14-16], in the Bethe approximation [17] and using the renormalization group method [18]. Phase separation has been studied in dilute magnets in the mean field approximation [9], and in systems with both magnetic components in the mean field and Bethe approximations [19]. The coexistence of structural order and phase separation and the possibility of phase transitions from the ordered to the phase separated system induced by a change of temperature and concentration have been studied for an FCC lattice by Rossiter and Lawrence in the Bragg-Williams approximation [20] and with the cluster variation method [21]. As follows from this review, most of the quoted works were devoted to structural order brought about by the presence of magnetic atoms. Phase separation as well as the coexistence of different ordered states and structural phase transitions from the ordered to separated system which are characteristic of some compounds like Fe-AI, Fe-Si and Fe-Ni attracted much less attention. In this paper we are going to discuss this phenomenon. We consider a dilute magnet with a BCC structure divided into 4 sublattices in which the interactions are confined to the first and second coordination spheres. Besides the exchange interactions we take into account the chemical interactions between the first neighbours which tend to order or to separate the atoms. In this way, depending on the signs of the coupling parameters two pairs of competing interactions in the system can appear. The consequence of this fact is the appearance of spontaneous metamagnetic as well as re-entrant transitions. All possible phases at zero temperature have been determined in part I of this paper, the temperature-concentration phase diagrams are the subject of the forthcoming part II.
2. Model
The subject of our consideration is a dilute magnet composed of N A magnetic atoms, of type A and N B nonmagnetic atoms of type B, saturating N sites of a BCC lattice, g A q- N B = N .
(1)
Two types of interactions are taken into account: a magnetic interaction between the first and second neighbours with exchange integrals 11 and 12 respectively and a nonmagnetic interaction (also known as chemical or structural) between the first neighbours only described by the potentials V AA, V BB and V AB. The atoms are allowed to migrate freely in the lattice taking a spatial
H . Jarz¢cka a n d B . F e c h n e r / M a g n e t i c a n d structural p h a s e transitions
159
distribution determined by the thermodynamic equilibrium conditions (annealed model). The Hamiltonian of the system is composed of a part describing chemical interactions Y(o-d ( o - d means order-disorder) and a part describing magnetic interactions ~mag" The nonmagnetic part of the Hamiltonian is assumed to be of the form 1
.. lVij x i x j +vi~ x i x j + v i i t,x i x j +XBixA)],
(2)
q
where x~ is an occupation operator of an ith site with an atom of type h (h = A, B). This operator takes the value 1 when an atom h is at the site i and equals 0 when this site is occupied by an atom of the second type. Since the system is saturated and the same site cannot be populated with two atoms at the same time, the operators satisfy the relations A
B
A
X i ~- X i = i ,
Xi
B
xi
(3)
= 0.
Depending on the sign of the expression V = V A A -}- V BB - 2 V A B the chemical interactions alone induce structural ordering of the system for V > 0, or result in a phase separation for V < 0. Structural ordering is evidenced as different occupation of crystalline sublattices, which means that one of the sublattices is homogeneously populated by a greater number of h type atoms, than the other, thus: A
A
A
P-1 = P-2 = P-
A
'
A
A
Pl31 = Pa2 = Pa '
A>
p,
A
pa ,
(4)
W here
p ,A is the probability of a v sublattice population with h type atoms. Such a distribution corresponds to the superstructure B2. The other superstructures B32 and DO 3 which may be formed in a BCC lattice are not energetically favorable because the chemical interactions were taken only between the first neighbours. When at T = 0 the system is structurally ordered and N A ~< N B, all A type atoms populate only the ot sublattice and then A
p~ = 2 p ,
(5)
where p = NA/N is the concentration of A type atoms (fig. 1). When N A > N B, the whole ot sublattice is populated only with A atoms with the probability p g _-- 1. With increasing temperature the atoms may migrate freely in the lattice tending to occupy all sites with the same probability. The system is then structurally disordered.
160
H. Jarzqcka and B. Fechner / Magnetic and structural phase transitions
~2
'£2
'CI •
00,
IF
OP2
0#1
j'Z
d
0~'
.c2
J
,q •
*(1
~2
~2
Fig. 1. Structural ordering at T = 0 for N A = N B. • A type atoms; © B type atoms, ai, 13j denote the sites of the four sublattices i, j = 1, 2.
The phase separation (fig. 2) is a spatial distribution of atoms which induces the appearance of two different regions (or phases), ~ and B, differing in the concentration of A type atoms. If these atoms occupy the phase p~ (IX = ~, B) with the probability pAp~ then the phases 7 and B satisfy the relation pA~ > pAa
Thus, if at T = 0 the phase separation takes place, A type atoms gathered in one block form the phase 7 with pA'v
-----
1,
pA~ = 0,
(6)
and, when T is different from 0, the atoms may migrate in the lattice which means that p g ~ ] , p A ~ 0 " Of course the phases may shrink or expand according to the equilibrium conditions. The magnetic part of the Hamiltonian is assumed to be ~mag = -
E .t i.j.a. i. J j x iAx jA -
gl~aH
~
ij
(7)
S~x A
i
~2
'q
'£1
@.~1
~C2
@,B2
L
'£1
0.~
0#2
,i
r~
"2
phase
aC2
':1
d phase
Fig. 2. Phase separation at T = 0.
C2
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
161
where H is the magnetic field; Iq = 11 when i, j is a pair of the first neighbours, Iq = I 2 when i, j is a pair of second neighbours and Iij = 0 in other cases. In a BCC lattice saturated with only magnetic atoms the following types of magnetic ordering are possible depending on the values and signs of both exchange integrals: (i) ferromagnetic ordering (F), (ii) antiferromagnetic orderings of first (AF 1) and second (AF2) kind, when the magnetic moments of the first and second neighbours, respectively, are antiparallel. When a number of sites are populated by nonmagnetic atoms, the system becomes a dilute magnet. Magnetic ineractions are not only responsible for producing a magnetically ordered state but they also affect the atomic distribution. If at T-- 0 the magnetic interactions were acting alone, they would cause phase separation.
3. Order parameters Taking into account both parts of the Hamiltonian (2) and (7) we may expect that magnetic interactions will influence the structural state of the system as well as the other way round, nonmagnetic interactions will affect the type of magnetic ordering through a change of the structural distribution. In order to describe the aforementioned magnetic and structural states we distinguished 4 sublattices Otl, or2, 131, 62 in the BCC lattice (fig. 1). Let p ~ be the probability of finding a h type atom in the v sublattice being a part of the tx phase; then
(8) where i is an arbitrary site which belongs to the phase ix and the sublattice v and ( . . . ) denotes a thermal average taken for a fixed number N g of magnetic atoms. The probabilities p ~ and pX~ satisfy the relations
p ~ . _ 4N~ ¢ N"
'
pX~_
~],, N~ ¢ N~ ,
(9)
where h = A, B; ix = -y, 8; and v = Otl, or2, 61, 62" N " / 4 is the number of sites in the phase ix which belong to the same sublattice, and N~ ¢ is the number of h
H. Jarzfcka and B. Fechner / Magnetic and structural phase transitions
162
type atoms in the phase ix and occupying the sublattice v. This means that I [ _hm
pX,
= 4 I" ] J a l
k~
(lO)
k~
P~2 )
4- ~t-~2 nXv" + Pl31 +
Thus a structural state of the system is characterized by 18 parameters: 16 probabilities p~" and two numbers N ~. These parameters are interrelated through the following 10 dependences: 1 ~
A~,
AV
A,
=p
(11)
N v + N ~= N,
(12)
pA¢ + p ~
(13)
= 1,
which are a consequence of the facts that the concentration of A type atoms is fixed and that the lattice is saturated with atoms. Because of the saturation of the lattice the index A in all the parameters can be omitted. This means that N B
Xi
= l--
m
N~"-
xi '
-
II, -
4
--
N~ , (14)
= 1-
p:
pB~
,
=
1 -p~ .
As we took into account only the structural interactions between the first neighbours there would be no mechanism causing different distributions of the same kind of atoms within the same phase between the sublattices a l and ot2 as well as 131 and 132. This leads to the four additional relations -
P~2
= P~
'
ix = ~, 8. P ~ l ---
~ = pt32
p~
(15)
,
Therefore, for unambiguous description of the structural state of the system we need only 4 parameters which will assume the role of order parameters. They are
Nv 0 = -N
(16)
describing the size of the ~/phase, m = pV - pa
(17)
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
163
describing the difference in population of ~/and 8 phases by magnetic atoms, where pV and pS are the probabilities of finding a magnetic atom in the phases ~/ and 8, respectively, and two parameters describing structural ordering within a given phase, L ~ =p~-p~,
Ix ='y, 8.
(18)
With the help of them we can describe the following sructural states. I. Phase separation
L ~ = L s = 0.
(19)
At T = 0 , 0 = p and m = 1. With increasing temperature the phase size changes: when 0---> 1 the phase ~/increases, until it eventually fills the whole system, when 0 = 0 it disappears. A change in phase size may be accompanied by a change in population of phases with magnetic atoms. This is described by the parameter m. H. Structural order
The system is structurally ordered when the order parameters fulfill the following relations: a) m = 0, L v= L ~= L, which means that the ~ and 8 phases are identical; b) 0 = 1, L v -- L, only the ~/phase is in the system; and
(20)
c) 0 = 0, L ~= L, only the 8 phase is in the system. In all those cases at T = 0 /2p
for p~< ½ ,
(21)
L 2(l-p)
for p/> ½.
With increasing temperature L tends to zero. 111. A mixed phase
This is a state with two phases ~/and 8, one of which is characterized by an unhomogeneous population of sublattices. A state like that requires that L~0,
m~0,
0~0,
0~1.
(22)
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
164
IV. Structural disorder
In this state both sublattices ~ and 13 are occupied by A atoms with the same probability, p~ = p . It is described by one of the following conditions: a)
L ~=L ~=0,
m=0,
b)
0 = 1,
L v =0,
c)
0 = 0,
L ~ = 0.
(23)
The probabilities of finding magnetic atoms in individual phases and sublattices can be expressed in terms of the order parameters as follows: p]=p+m(1-0)+
½L v,
p~ = p - mO + ½L ~ ,
p ; = p + m ( 1 - 0 ) - ½L ~ , p~=p-mO-½L
~.
(24)
A magnetic state of the system may be described by specifying magnetizations of all sublattices within a given phase: z
p~ Or v - -
(Sixi)
S
'
P~= % ~'
v = or1, ~2,131,132,
(25)
where i is a site of the phase ix in the sublattice v. In the most general case the number of order parameters is 12. A full description of a macroscopic state of the system requires a set of 12 nonlinear equations to be solved. In the other structural states some of the order parameters disappear and so the number of equations to be solved is reduced. To simplify the problem we decided to exclude the mixed phase from our considerations. 4. The ground state The calculations were performed by the effective field method with the higher order terms decoupled as follows:
= (STxi)
z
) ,
(26) (xixj) :
.
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
165
T h e internal e n e r g y p e r site calculated for H = 0 is u = -¼[0E
+ (1 - 0 ) E
1((L )2 -
+ 1 g Z l { O [ m 2 ( 1 - O) -
+pZ _ (1LS)2} ,
(27)
where EIx = I 1 z 1 5 2 ( o - ~ 1
Ix "b 2 1 2 z 2 S 2 ( O - ~ 1 0 - ~ 2 -~- O-Ix a 2 )\["o - 1 3Ix 1 + O-[32)
+ O-1310-[32) Ix Ix
and z k is the n u m b e r o f sites in the k t h c o o r d i n a t i o n sphere. F r o m the m i n i m u m conditions for U we f o u n d the following g r o u n d states: - Structurally s e p a r a t e d ferro or a n t i f e r r o m a g n e t (X, s), X = F, A F 1 or A F 2, with the o r d e r p a r a m e t e r s m = 1, 0 = p ,
Io-~l=l,
O-~=o,
v
where
= 0 / . 1 , 0 / , 2 , 1~1, [~2 •
- Structurally o r d e r e d ferro, ferri or a n t i f e r r o m a g n e t (X, o), X = F, FI, or A F 2, with the structural o r d e r p a r a m e t e r s fulfilling one of the relations (20) and
=f2p, L
1% 1=2p,
O-[3,= 0
for p~< 1 ,
1%/1=1,
1%,1=2p-1
forp~>~,
/ [2(l-p),
i=1,2,
w h e r e FI stands for a f e r r i m a g n e t with the nearest neighbours o r d e r e d antiferromagnetically. - Structurally d i s o r d e r e d ferro or a n t i f e r r o m a g n e t (X, d), X = F or A F 1 , with structural o r d e r p a r a m e t e r s satisfying o n e of the conditions (23). T h e m a g n e t i z a t i o n s d e p e n d on the m a g n e t i c state X as follows. For X = F,
O-Ix ~ ct I
Ix ~ O'ct2
Ix O'ct ,
O'Ix ~- O'Ix ~--- Ix [31 [32 0"[3 ~.
Ix sgn o -Ix = sgn 0-[3. F o r X = AF~ o r F I ,
IX ~ O'c'1
O'IX ~ ~2
Ix O-~ '
O'Ix ~ 131
O'Ix ~ 17'2
Ix O-[3 '
or [31 l-t ~
- - O r I~ [32 "
Ix . sgn O-, = - s g n O-~ For X = AF 2 ,
IX ~r~l
~
- - O -p" a2 '
W h e n 11 = 12 = 0 the g r o u n d state d e p e n d s on the sign o f V and b e c o m e s - a structurally o r d e r e d p a r a m a g n e t (P, o) for V > 0 or - a structurally s e p a r a t e d p a r a m a g n e t (P, s) for V < 0.
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H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
The regions of existence of particular phases for V > 0 versus the concentration of magnetic atoms as well as the parameters Jx=I~/IVI, J2= I2z2/(IvIzl) are shown in fig. 3. As follows, for fixed J1 and J2 and V > 0 the ground state may change depending on p. Regions in the J~, J2 plane for which phase changes with increasing p can be determined are shown in fig. 4. When exchange integrals come from the regions (1)-(5) the ground state is
8
l J2 /
(AFvs) ~
(F,s)
|
/(AF2,o)x (A~,s)
C
(AFI,s) / ~
(F,s)
CAF~,s) tFI,0 t F . o )
(AF2~ (AF21s) Fig. 3. The ground state versus magnetic atoms concentration (b) 0
for the potential
V > 0. (a) p = 0,
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
d
I
167
kJ2
(AFt,s)i
I
(F,s) (F,o)\
(F,s)
(AFt,s)
(AF20),
(AE s)
(AF 2s) Fig. 3. (cont.).
independent ofp. If they are great when compared with V (regions (1) and (2)) the actual kind of magnetic and structural ordering is determined by magnetic interactions which create the structure that would occur in the system without chemical interactions, i.e. (F, s) and (AF1, s), respectively. If J~, J: come from the regions (3), (4), (5) the structural state is determined by chemical interactions and its magnetic ordering is such as for p = 1. In the sequence of increasing numbers these are the states (AF2, o), (F, o), (FI, o). For exchange integrals from the other regions ((6)-(14)) an increase in magnetic atoms concentration may induce one of the following changes in the ground state: (i) discontinuous change in structural ordering, (ii) discontinuous change in magnetic ordering, (iii) simultaneous discontinuous change in magnetic and structural orderings.
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
168
J2
121
{ill
17)
1|)
11~1
Fig. 4. Regions in the J,, J2 plane for which phase changes with increasing p can be determined.
(i) Discontinuous changes in the structural ordering with the magnetic ordering staying the same. A structurally separated system (X, s), with increasing concentration p, undergoes a discontinuous change into the state of structural ordering (X, o) and then comes back to the (X, s) state. Schematically it may be written down as P
p
(x, s)----. (x, o)
, (x, s).
Such transitions occur for the coupling parameters from the regions (6), (7) and (14). In particular, P
(F, o)
P
in (6)
(F, s) ~
~ (F, s),
in (7)
(AWl ' s) ~ P
in (14)
(AF2 ' s) ~ P (AF 2, o ) ~
(FI, o) ~ P
(AF,, s), (AF 2 , s) •
(ii) Discontinuous changes in the magnetic ordering with the structural ordering staying the same, in (8)
(AF2, o)
in (9)
(AF 2, o)
p
P
, (F, o ) , , (FI, o).
H. Jarzecka and B. Fechner / Magnetic and structural phase transitions
169
(iii) Simultaneous discontinuous changes in magnetic and structural orderings. They occur for coupling parameters from the regions (10)-(13). These are the following transitions: in (10)
(AF2, o ) ~ P
in (11)
(AF2, o)
in (12)
(F, s)
in (13)
(AF 1, s) ~
p
p
(F, d ) , , (AF1, d ) ,
, (AF 2, o)
p
, (F, s),
(AF 2, o) ~
(AF1, s).
The ground state dependence on the potential V at the fixed concentration p, for p < ½, is shown in fig. 5. For p ~> ½ the character of the dependence is similar in the sense that with decreasing V the regions (s) expand to fill the whole plane for V ~<0. As follows from fig. 5c the magnetic interactions alone (V= 0) lead to phase separation with the same magnetic ordering as of a
a
(AS,s) /
(~s}
(AF~js) b
C
(AS s)
(F,s)
(A~,s)
(F,s)
Fig. 5. The ground state versus the potential V for p~< ½ (a) V = Va, Va > 0 ;
o
k.
(b) V = V b ,
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H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
system saturated with magnetic atoms. For V < 0 both types of interactions, magnetic and chemical, act in the same direction stabilizing each other.
5. Conclusions As follows from the above analysis, the ground state strongly depends on the interrelations a m o n g the coupling p a r a m e t e r s J1, J2, V (V >0), and the concentration p. T h e actual state of the system is a result of the competition between the magnetic and chemical interactions, which m a y lead to the following situations. (i) Magnetic interactions m a y change the type of structural ordering f r o m (o) to (s) or lead to a structural disorder (d). (ii) Chemical interactions in the case of inhomogeneous population of the sublattices are responsible for differentiation of their magnetizations I ol and even m a y change the magnetic ordering. The mechanism of the change in magnetic ordering is easily understood as follows. At T --- 0, due to chemical interactions, magnetic atoms for p ~< 1 may occupy only the a sublattice, and for p > ½, N A - N / 2 atoms will populate the [3 sublattice. In both cases the n u m b e r of pairs of magnetic atoms which are the second neighbours will dominate over the n u m b e r of first neighbours pairs. Thus, the type of magnetic ordering will be determined by the second neighbour interactions which produce the A F 2 state, instead of F or AF1, as it would have been for V = 0.
Acknowledgement This work has b e e n supported by the Institute of Physics of the Polish A c a d e m y of Sciences under Project No C P B P 01.04-I1.1.4.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
V. Pierron-Bohnes, M.C. Cadeville and F. Gautier, J. Phys. F 13 (1983) 1689. V. Pierron-Bohnes, M.C. Cadeville and G. Parette, J. Phys. F 15 (1985) 1441. I. Mirabeau, M.C. Cadeville, G. Parette and I.A. Campbell, J. Phys. F 12 (1982) 25. V. Pierron-Bohnes, I. Mirabeau, E. Balanzat and M.C. Cadeville, J. Phys. F 14 (1984) 197. C.R. Houska, J. Phys. Chem. Solids 24 (1963) 95. J.S. Kouvel, Magnetism and Metallurgy (Academic Press, New York, 1969). C.E. Violet and R.J. Borg, Phys. Rev. Lett. 51 (1983) 1073. S. Katano and M. Lizumi, Phys. Rev. Lett. 52 (1984) 835. J. Urias and J.L. Moran-Lopez, Phys. Rev. B 26 (1982) 2669.
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G.M. Bell and D.A. Lavis, Phil. Mag. 11 (1965) 937. D.A. Lavis and G.M. Bell, Phil. Mag. 15 (1967) 587. D.A. Lavis and W.M. Fairbairn, Phil. Mag. 13 (1966) 477. T. Kawasaki, Prog. Theor. Phys. 58 (1977) 1357. R.A. Tahir-Kheli and T. Kawasaki, J. Phys. C 10 (1977) 2207. J.L. Moran-Lopez and L.M. Falicov, Solid State Commun. 31 (1979) 325. J.L. Moran-Lopez and L.M. Falicov, J. Phys. C 13 (1980) 1715. F.J. Martinez-Herrera, F.J. Mejira-Lira, F.J. Aguilera-Granja and J.L. Moran-Lopez, Phys. Rev. B 31 (1985) 1686. Z. Racz and M. Collins, Phys. Rev. B 21 (1980) 229. M. Yamashita and H. Nakano, Prog. Theor. Phys. 61 (1979) 764. P.L. Rossiter and P.J. Lawrence, Phil. Mag. A4 (1984) 535. P.L. Rossiter and P.J. Lawrence, Mat. Res. Soc. Symp. Proc. 21 (1984) 481.
MAGNETIC AND STRUCTURAL PHASE TRANSITIONS IN A DILUTE M A G N E T W I T H CONFIGURATIONAL DEGREES OF F R E E D O M I. THE GROUND STATE
H. JARZI~CKA and B. F E C H N E R Institute of Physics, Adam Mickiewicz University, Poznati, Poland
Received 22 November 1989 An annealed binary system with one magnetic component is considered. Chemical and magnetic interactions are taken into account which for some signs of their parameters induce two kinds of competitions between the magnetic atoms. The ground state versus the concentration of magnetic atoms in the mean field approximation is discussed. Characteristic diagrams are presented exposing the connection between magnetic and structural orders.
I. I n t r o d u c t i o n
The interrelation between magnetism and spatial distribution of atoms in a lattice has been investigated in many alloys and compounds. The best known compounds showing structural order are those of the types F e - C o [1, 2], F e - V [3, 4], F e - A I [5] crystallizing in the BCC lattice and F e - N i , F e - P d , F e - P t , M n - N i [6] crystallizing in the FCC lattice. On the other hand, a tendency to phase separation was observed in the alloys A u - F e [7], F e - C r [8], C u - N i , A u - N i , C u - P t [9]. A n o t h e r group of compounds including such alloys as Fe-A1, F e - N i may be structurally ordered or separated depending on the temperature and concentration of one of their components. Rich experimental evidence resulted in many theoretical works, which may be divided into three groups: those devoted to the studies of structural order, to phase separation and those studying the possibilities of coexistence of structural order and phase separation. At first structural order has been studied in dilute binary alloys with one magnetic component in the mean field approximation [5, 10, 11], in the Bethe approximation [12] and using the G r e e n function m e t h o d [13]. Then it has been studied in binary systems of two magnetic components in the mean field 0378-4371/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
158
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
approximation [1, 2, 14-16], in the Bethe approximation [17] and using the renormalization group method [18]. Phase separation has been studied in dilute magnets in the mean field approximation [9], and in systems with both magnetic components in the mean field and Bethe approximations [19]. The coexistence of structural order and phase separation and the possibility of phase transitions from the ordered to the phase separated system induced by a change of temperature and concentration have been studied for an FCC lattice by Rossiter and Lawrence in the Bragg-Williams approximation [20] and with the cluster variation method [21]. As follows from this review, most of the quoted works were devoted to structural order brought about by the presence of magnetic atoms. Phase separation as well as the coexistence of different ordered states and structural phase transitions from the ordered to separated system which are characteristic of some compounds like Fe-AI, Fe-Si and Fe-Ni attracted much less attention. In this paper we are going to discuss this phenomenon. We consider a dilute magnet with a BCC structure divided into 4 sublattices in which the interactions are confined to the first and second coordination spheres. Besides the exchange interactions we take into account the chemical interactions between the first neighbours which tend to order or to separate the atoms. In this way, depending on the signs of the coupling parameters two pairs of competing interactions in the system can appear. The consequence of this fact is the appearance of spontaneous metamagnetic as well as re-entrant transitions. All possible phases at zero temperature have been determined in part I of this paper, the temperature-concentration phase diagrams are the subject of the forthcoming part II.
2. Model
The subject of our consideration is a dilute magnet composed of N A magnetic atoms, of type A and N B nonmagnetic atoms of type B, saturating N sites of a BCC lattice, g A q- N B = N .
(1)
Two types of interactions are taken into account: a magnetic interaction between the first and second neighbours with exchange integrals 11 and 12 respectively and a nonmagnetic interaction (also known as chemical or structural) between the first neighbours only described by the potentials V AA, V BB and V AB. The atoms are allowed to migrate freely in the lattice taking a spatial
H . Jarz¢cka a n d B . F e c h n e r / M a g n e t i c a n d structural p h a s e transitions
159
distribution determined by the thermodynamic equilibrium conditions (annealed model). The Hamiltonian of the system is composed of a part describing chemical interactions Y(o-d ( o - d means order-disorder) and a part describing magnetic interactions ~mag" The nonmagnetic part of the Hamiltonian is assumed to be of the form 1
.. lVij x i x j +vi~ x i x j + v i i t,x i x j +XBixA)],
(2)
q
where x~ is an occupation operator of an ith site with an atom of type h (h = A, B). This operator takes the value 1 when an atom h is at the site i and equals 0 when this site is occupied by an atom of the second type. Since the system is saturated and the same site cannot be populated with two atoms at the same time, the operators satisfy the relations A
B
A
X i ~- X i = i ,
Xi
B
xi
(3)
= 0.
Depending on the sign of the expression V = V A A -}- V BB - 2 V A B the chemical interactions alone induce structural ordering of the system for V > 0, or result in a phase separation for V < 0. Structural ordering is evidenced as different occupation of crystalline sublattices, which means that one of the sublattices is homogeneously populated by a greater number of h type atoms, than the other, thus: A
A
A
P-1 = P-2 = P-
A
'
A
A
Pl31 = Pa2 = Pa '
A>
p,
A
pa ,
(4)
W here
p ,A is the probability of a v sublattice population with h type atoms. Such a distribution corresponds to the superstructure B2. The other superstructures B32 and DO 3 which may be formed in a BCC lattice are not energetically favorable because the chemical interactions were taken only between the first neighbours. When at T = 0 the system is structurally ordered and N A ~< N B, all A type atoms populate only the ot sublattice and then A
p~ = 2 p ,
(5)
where p = NA/N is the concentration of A type atoms (fig. 1). When N A > N B, the whole ot sublattice is populated only with A atoms with the probability p g _-- 1. With increasing temperature the atoms may migrate freely in the lattice tending to occupy all sites with the same probability. The system is then structurally disordered.
160
H. Jarzqcka and B. Fechner / Magnetic and structural phase transitions
~2
'£2
'CI •
00,
IF
OP2
0#1
j'Z
d
0~'
.c2
J
,q •
*(1
~2
~2
Fig. 1. Structural ordering at T = 0 for N A = N B. • A type atoms; © B type atoms, ai, 13j denote the sites of the four sublattices i, j = 1, 2.
The phase separation (fig. 2) is a spatial distribution of atoms which induces the appearance of two different regions (or phases), ~ and B, differing in the concentration of A type atoms. If these atoms occupy the phase p~ (IX = ~, B) with the probability pAp~ then the phases 7 and B satisfy the relation pA~ > pAa
Thus, if at T = 0 the phase separation takes place, A type atoms gathered in one block form the phase 7 with pA'v
-----
1,
pA~ = 0,
(6)
and, when T is different from 0, the atoms may migrate in the lattice which means that p g ~ ] , p A ~ 0 " Of course the phases may shrink or expand according to the equilibrium conditions. The magnetic part of the Hamiltonian is assumed to be ~mag = -
E .t i.j.a. i. J j x iAx jA -
gl~aH
~
ij
(7)
S~x A
i
~2
'q
'£1
@.~1
~C2
@,B2
L
'£1
0.~
0#2
,i
r~
"2
phase
aC2
':1
d phase
Fig. 2. Phase separation at T = 0.
C2
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
161
where H is the magnetic field; Iq = 11 when i, j is a pair of the first neighbours, Iq = I 2 when i, j is a pair of second neighbours and Iij = 0 in other cases. In a BCC lattice saturated with only magnetic atoms the following types of magnetic ordering are possible depending on the values and signs of both exchange integrals: (i) ferromagnetic ordering (F), (ii) antiferromagnetic orderings of first (AF 1) and second (AF2) kind, when the magnetic moments of the first and second neighbours, respectively, are antiparallel. When a number of sites are populated by nonmagnetic atoms, the system becomes a dilute magnet. Magnetic ineractions are not only responsible for producing a magnetically ordered state but they also affect the atomic distribution. If at T-- 0 the magnetic interactions were acting alone, they would cause phase separation.
3. Order parameters Taking into account both parts of the Hamiltonian (2) and (7) we may expect that magnetic interactions will influence the structural state of the system as well as the other way round, nonmagnetic interactions will affect the type of magnetic ordering through a change of the structural distribution. In order to describe the aforementioned magnetic and structural states we distinguished 4 sublattices Otl, or2, 131, 62 in the BCC lattice (fig. 1). Let p ~ be the probability of finding a h type atom in the v sublattice being a part of the tx phase; then
(8) where i is an arbitrary site which belongs to the phase ix and the sublattice v and ( . . . ) denotes a thermal average taken for a fixed number N g of magnetic atoms. The probabilities p ~ and pX~ satisfy the relations
p ~ . _ 4N~ ¢ N"
'
pX~_
~],, N~ ¢ N~ ,
(9)
where h = A, B; ix = -y, 8; and v = Otl, or2, 61, 62" N " / 4 is the number of sites in the phase ix which belong to the same sublattice, and N~ ¢ is the number of h
H. Jarzfcka and B. Fechner / Magnetic and structural phase transitions
162
type atoms in the phase ix and occupying the sublattice v. This means that I [ _hm
pX,
= 4 I" ] J a l
k~
(lO)
k~
P~2 )
4- ~t-~2 nXv" + Pl31 +
Thus a structural state of the system is characterized by 18 parameters: 16 probabilities p~" and two numbers N ~. These parameters are interrelated through the following 10 dependences: 1 ~
A~,
AV
A,
=p
(11)
N v + N ~= N,
(12)
pA¢ + p ~
(13)
= 1,
which are a consequence of the facts that the concentration of A type atoms is fixed and that the lattice is saturated with atoms. Because of the saturation of the lattice the index A in all the parameters can be omitted. This means that N B
Xi
= l--
m
N~"-
xi '
-
II, -
4
--
N~ , (14)
= 1-
p:
pB~
,
=
1 -p~ .
As we took into account only the structural interactions between the first neighbours there would be no mechanism causing different distributions of the same kind of atoms within the same phase between the sublattices a l and ot2 as well as 131 and 132. This leads to the four additional relations -
P~2
= P~
'
ix = ~, 8. P ~ l ---
~ = pt32
p~
(15)
,
Therefore, for unambiguous description of the structural state of the system we need only 4 parameters which will assume the role of order parameters. They are
Nv 0 = -N
(16)
describing the size of the ~/phase, m = pV - pa
(17)
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
163
describing the difference in population of ~/and 8 phases by magnetic atoms, where pV and pS are the probabilities of finding a magnetic atom in the phases ~/ and 8, respectively, and two parameters describing structural ordering within a given phase, L ~ =p~-p~,
Ix ='y, 8.
(18)
With the help of them we can describe the following sructural states. I. Phase separation
L ~ = L s = 0.
(19)
At T = 0 , 0 = p and m = 1. With increasing temperature the phase size changes: when 0---> 1 the phase ~/increases, until it eventually fills the whole system, when 0 = 0 it disappears. A change in phase size may be accompanied by a change in population of phases with magnetic atoms. This is described by the parameter m. H. Structural order
The system is structurally ordered when the order parameters fulfill the following relations: a) m = 0, L v= L ~= L, which means that the ~ and 8 phases are identical; b) 0 = 1, L v -- L, only the ~/phase is in the system; and
(20)
c) 0 = 0, L ~= L, only the 8 phase is in the system. In all those cases at T = 0 /2p
for p~< ½ ,
(21)
L 2(l-p)
for p/> ½.
With increasing temperature L tends to zero. 111. A mixed phase
This is a state with two phases ~/and 8, one of which is characterized by an unhomogeneous population of sublattices. A state like that requires that L~0,
m~0,
0~0,
0~1.
(22)
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
164
IV. Structural disorder
In this state both sublattices ~ and 13 are occupied by A atoms with the same probability, p~ = p . It is described by one of the following conditions: a)
L ~=L ~=0,
m=0,
b)
0 = 1,
L v =0,
c)
0 = 0,
L ~ = 0.
(23)
The probabilities of finding magnetic atoms in individual phases and sublattices can be expressed in terms of the order parameters as follows: p]=p+m(1-0)+
½L v,
p~ = p - mO + ½L ~ ,
p ; = p + m ( 1 - 0 ) - ½L ~ , p~=p-mO-½L
~.
(24)
A magnetic state of the system may be described by specifying magnetizations of all sublattices within a given phase: z
p~ Or v - -
(Sixi)
S
'
P~= % ~'
v = or1, ~2,131,132,
(25)
where i is a site of the phase ix in the sublattice v. In the most general case the number of order parameters is 12. A full description of a macroscopic state of the system requires a set of 12 nonlinear equations to be solved. In the other structural states some of the order parameters disappear and so the number of equations to be solved is reduced. To simplify the problem we decided to exclude the mixed phase from our considerations. 4. The ground state The calculations were performed by the effective field method with the higher order terms decoupled as follows:
= (STxi)
z
) ,
(26) (xixj) :
.
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
165
T h e internal e n e r g y p e r site calculated for H = 0 is u = -¼[0E
+ (1 - 0 ) E
1((L )2 -
+ 1 g Z l { O [ m 2 ( 1 - O) -
+pZ _ (1LS)2} ,
(27)
where EIx = I 1 z 1 5 2 ( o - ~ 1
Ix "b 2 1 2 z 2 S 2 ( O - ~ 1 0 - ~ 2 -~- O-Ix a 2 )\["o - 1 3Ix 1 + O-[32)
+ O-1310-[32) Ix Ix
and z k is the n u m b e r o f sites in the k t h c o o r d i n a t i o n sphere. F r o m the m i n i m u m conditions for U we f o u n d the following g r o u n d states: - Structurally s e p a r a t e d ferro or a n t i f e r r o m a g n e t (X, s), X = F, A F 1 or A F 2, with the o r d e r p a r a m e t e r s m = 1, 0 = p ,
Io-~l=l,
O-~=o,
v
where
= 0 / . 1 , 0 / , 2 , 1~1, [~2 •
- Structurally o r d e r e d ferro, ferri or a n t i f e r r o m a g n e t (X, o), X = F, FI, or A F 2, with the structural o r d e r p a r a m e t e r s fulfilling one of the relations (20) and
=f2p, L
1% 1=2p,
O-[3,= 0
for p~< 1 ,
1%/1=1,
1%,1=2p-1
forp~>~,
/ [2(l-p),
i=1,2,
w h e r e FI stands for a f e r r i m a g n e t with the nearest neighbours o r d e r e d antiferromagnetically. - Structurally d i s o r d e r e d ferro or a n t i f e r r o m a g n e t (X, d), X = F or A F 1 , with structural o r d e r p a r a m e t e r s satisfying o n e of the conditions (23). T h e m a g n e t i z a t i o n s d e p e n d on the m a g n e t i c state X as follows. For X = F,
O-Ix ~ ct I
Ix ~ O'ct2
Ix O'ct ,
O'Ix ~- O'Ix ~--- Ix [31 [32 0"[3 ~.
Ix sgn o -Ix = sgn 0-[3. F o r X = AF~ o r F I ,
IX ~ O'c'1
O'IX ~ ~2
Ix O-~ '
O'Ix ~ 131
O'Ix ~ 17'2
Ix O-[3 '
or [31 l-t ~
- - O r I~ [32 "
Ix . sgn O-, = - s g n O-~ For X = AF 2 ,
IX ~r~l
~
- - O -p" a2 '
W h e n 11 = 12 = 0 the g r o u n d state d e p e n d s on the sign o f V and b e c o m e s - a structurally o r d e r e d p a r a m a g n e t (P, o) for V > 0 or - a structurally s e p a r a t e d p a r a m a g n e t (P, s) for V < 0.
166
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
The regions of existence of particular phases for V > 0 versus the concentration of magnetic atoms as well as the parameters Jx=I~/IVI, J2= I2z2/(IvIzl) are shown in fig. 3. As follows, for fixed J1 and J2 and V > 0 the ground state may change depending on p. Regions in the J~, J2 plane for which phase changes with increasing p can be determined are shown in fig. 4. When exchange integrals come from the regions (1)-(5) the ground state is
8
l J2 /
(AFvs) ~
(F,s)
|
/(AF2,o)x (A~,s)
C
(AFI,s) / ~
(F,s)
CAF~,s) tFI,0 t F . o )
(AF2~ (AF21s) Fig. 3. The ground state versus magnetic atoms concentration (b) 0
for the potential
V > 0. (a) p = 0,
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
d
I
167
kJ2
(AFt,s)i
I
(F,s) (F,o)\
(F,s)
(AFt,s)
(AF20),
(AE s)
(AF 2s) Fig. 3. (cont.).
independent ofp. If they are great when compared with V (regions (1) and (2)) the actual kind of magnetic and structural ordering is determined by magnetic interactions which create the structure that would occur in the system without chemical interactions, i.e. (F, s) and (AF1, s), respectively. If J~, J: come from the regions (3), (4), (5) the structural state is determined by chemical interactions and its magnetic ordering is such as for p = 1. In the sequence of increasing numbers these are the states (AF2, o), (F, o), (FI, o). For exchange integrals from the other regions ((6)-(14)) an increase in magnetic atoms concentration may induce one of the following changes in the ground state: (i) discontinuous change in structural ordering, (ii) discontinuous change in magnetic ordering, (iii) simultaneous discontinuous change in magnetic and structural orderings.
H. Jarz(cka and B. Fechner / Magnetic and structural phase transitions
168
J2
121
{ill
17)
1|)
11~1
Fig. 4. Regions in the J,, J2 plane for which phase changes with increasing p can be determined.
(i) Discontinuous changes in the structural ordering with the magnetic ordering staying the same. A structurally separated system (X, s), with increasing concentration p, undergoes a discontinuous change into the state of structural ordering (X, o) and then comes back to the (X, s) state. Schematically it may be written down as P
p
(x, s)----. (x, o)
, (x, s).
Such transitions occur for the coupling parameters from the regions (6), (7) and (14). In particular, P
(F, o)
P
in (6)
(F, s) ~
~ (F, s),
in (7)
(AWl ' s) ~ P
in (14)
(AF2 ' s) ~ P (AF 2, o ) ~
(FI, o) ~ P
(AF,, s), (AF 2 , s) •
(ii) Discontinuous changes in the magnetic ordering with the structural ordering staying the same, in (8)
(AF2, o)
in (9)
(AF 2, o)
p
P
, (F, o ) , , (FI, o).
H. Jarzecka and B. Fechner / Magnetic and structural phase transitions
169
(iii) Simultaneous discontinuous changes in magnetic and structural orderings. They occur for coupling parameters from the regions (10)-(13). These are the following transitions: in (10)
(AF2, o ) ~ P
in (11)
(AF2, o)
in (12)
(F, s)
in (13)
(AF 1, s) ~
p
p
(F, d ) , , (AF1, d ) ,
, (AF 2, o)
p
, (F, s),
(AF 2, o) ~
(AF1, s).
The ground state dependence on the potential V at the fixed concentration p, for p < ½, is shown in fig. 5. For p ~> ½ the character of the dependence is similar in the sense that with decreasing V the regions (s) expand to fill the whole plane for V ~<0. As follows from fig. 5c the magnetic interactions alone (V= 0) lead to phase separation with the same magnetic ordering as of a
a
(AS,s) /
(~s}
(AF~js) b
C
(AS s)
(F,s)
(A~,s)
(F,s)
Fig. 5. The ground state versus the potential V for p~< ½ (a) V = Va, Va > 0 ;
o
k.
(b) V = V b ,
170
H. Jarz¢cka and B. Fechner / Magnetic and structural phase transitions
system saturated with magnetic atoms. For V < 0 both types of interactions, magnetic and chemical, act in the same direction stabilizing each other.
5. Conclusions As follows from the above analysis, the ground state strongly depends on the interrelations a m o n g the coupling p a r a m e t e r s J1, J2, V (V >0), and the concentration p. T h e actual state of the system is a result of the competition between the magnetic and chemical interactions, which m a y lead to the following situations. (i) Magnetic interactions m a y change the type of structural ordering f r o m (o) to (s) or lead to a structural disorder (d). (ii) Chemical interactions in the case of inhomogeneous population of the sublattices are responsible for differentiation of their magnetizations I ol and even m a y change the magnetic ordering. The mechanism of the change in magnetic ordering is easily understood as follows. At T --- 0, due to chemical interactions, magnetic atoms for p ~< 1 may occupy only the a sublattice, and for p > ½, N A - N / 2 atoms will populate the [3 sublattice. In both cases the n u m b e r of pairs of magnetic atoms which are the second neighbours will dominate over the n u m b e r of first neighbours pairs. Thus, the type of magnetic ordering will be determined by the second neighbour interactions which produce the A F 2 state, instead of F or AF1, as it would have been for V = 0.
Acknowledgement This work has b e e n supported by the Institute of Physics of the Polish A c a d e m y of Sciences under Project No C P B P 01.04-I1.1.4.
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