Impurities in quasi one-dimensional systems : a study on
some thermodynamic properties
Citation for published version (APA):
Schouten, J. C. (1981). Impurities in quasi one-dimensional systems : a study on some thermodynamic
properties. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR71573
DOI:
10.6100/IR71573
Document status and date:
Published: 01/01/1981
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be
important differences between the submitted version and the official published version of record. People
interested in the research are advised to contact the author for the final version of the publication, or visit the
DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page
numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
providing details and we will investigate your claim.
IMPURITIES IN QUASI
ONE-DIMENSION AL
SYSTEMS
A STUDY ON SOME THERMODYNAMIC
PROPERTIES
J.
C. SCHOUTEN
IMPURITIES IN QUASI
ONE-DIMENSIONAL
SYSTEMS
A STUDY ONSOME THERMODYNAMIC
PROPERTIES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAADVAN DOCTOR
IN DE TECHNISCHE WETENSCHAPPEN AAN DE
TECHNISCHE HOGESCHOOL TE EINDHOVEN, OP
GEZAGVAN DE RECTOR MAGNIFICUS, PROF. IR.
J. ERKELENS, VOOR EEN COMMISSIE
AANGE-WEZEN DOOR HET COLLEGEVAN DEKANEN IN
HET OPENBAAR TE VERDEDIGEN OP DINSDAG
20 JANUARI 1981 TE 16.00 UUR
DOOR
JOSEPH CHRISTOPHORUS SCHOUTEN
Dit proefschrift is goedgekeurd door de promotoren Prof. Dr. ir. W.J.M. de Jonge en Prof. Dr. M.J. Steenland
"Alle mensen worden vrij
en rechten geboren. zijn
en geweten, en behoren e~kander in een
geest van broederschap te
"
Universele van de
Re eh ten van de 1.1ens, art. 1
Vastgesteld in de Algemene Vergadering van de Verenigde Naties op 10 december 1948
Table of Contents
I INTRODUCTION
II REVIEW OF RANDOM MAGNETIC SYSTEMS
2.1 Introduetion
2.2 Random magnetic systems
2.2.1 Introduetion
2.3
2.4
2.5
2.2.2 Amorphous magnetic systems 2.2.3 Regular random magnetic systems Two- and three-dimensional systems
2.3. I Introduetion
2.3.2 Dilute magnetic systems 2.3.3 Mixed magnetic systems
Quasi one-dimensional magnetic systems
2.4. I Introduetion
2.4.2 Mixed ld Ising systems 2.4.3 Mixed ld Heisenberg systems 2.4.4 Mixed 1d XY systems
Conclusion l i l EXPERIMENTAL METHODS
3.1 Introduetion
3.2 Preparation of the impure cyrstals
3.3 Chemica! analysis
3.4
3.5 3.6
Calorimetry
Thermometry and calibration
Susceptibi l i ty and magnetization
3.7 Dynamic susceptibility
IV COMPUTATIONAL METHODS FOR QUASI ONE-DIMENSIONAL DILUTE
SYSTEMS WITH ANISOTROPIC HEISENBERG INTERACTION
4.1 Introduetion
4.2 Transfer matrix formalism
4.3 Dilute anisatrapie chain systems
4 4 5 5 7 9 13 13 14 26 32 32 34 38 43 44 45 45 45 47 49 52 54 55 57 57 58 60
V VI I:HPURITIES IN Mn ++ SYSTE:HS 5.1 Introduetion 5.2 (CH 3)2NH2Mncl3 5. 2. I Structure 5.2.2 Specific heat 5.2.3 Susceptibi li ty 5.3 Impurity experiments 5.3.1 Introduetion 5.3.2 Three-dimensional ordering 5.3.3 :Hagnetic phase diagram 5.3.4 Susceptibility above TN 5.4 Concluding remarks
IMPURITIES IN Co++ SYSTE:HS
6.1 Introduetion
6.2 Some relevant properties of (CH3 ) 3NHCoC1 3 ·2H2
o
and CoC1 2 ·2(NC5H5)6.2.1 Trimethyl ammonium cabalt trichloride 6.2.2 Cabalt chloride dipyridine
6.3 Numerical calculation of TN(x) in.mixed quasi ld systems 6.4 Experimental results 69 69 71 71 72 78 83 83 83 87 92 99 100 100 101 101 103
VII THE :HIXED SYSTE:H CsMn(Cl 1_xBrx) 3 •2HzO
7.1 Introduetion 104 106 110 110 l i l 114 116 7.2 Some properties of pure C:HC and c~rn
7.3 Preparation of the samples 7.4 :Heasurements and discussion
References 122
Samenvatting 128
CHAPTER I
INTRODUCTION
The thermadynamie properties of systems containing
ÎQpurities have been investigated almast over the same period as the pure magnetic systems themselves. The reason for this seems at least twofold. First, the presence of impurities can hardly be avoided since real pure systems - without any biemishes - are not realized in nature, and secondly, the study of impure systems may yield additional information on tht pure system.
The effect of an impurity in a magnetic system will strongly depend on the character of the impurity, that is either magnetic or nonmag-netic, or alternatively, affecting the magnetic interaction or nat. The effect of an impurity also depends on the dimensionality of the system: in two-dimensional systems there will be less "ways around" the impurity than in three-dimensional systems, and in one-dimensional systems there is even no way around. It appears that the physical realization of such an one-dimensional system can be found in some magnetic crystals composed of long chains of magnetic ions with large intrachain interactions and very small interchain interactions (de Jongh and Miedema, 1974; Steiner et aZ., 1976; Mikeska, 1979).
At this point, it should be mentioned that the concepts "impurity" and "impure system", though commonly used, have not yet been well de-fined in the literature. These rather intuitive terms are nat very appropriate to describe the wide class of random magnetio systems, which are characterized, in one way or another, by a lack of regularity. In chapter II we will give a restricted review of the large variety of random magnetic systems, and the theoretica! and experimental studies which have been performed on them. It will appear that the systems described in this thesis belang to the so-called mixed magnetio systems, which are composed of two very similar regular systems of which at least one is magnetic.
As already mentioned above, one-dimensional (1d) magnetic systems form a special category because there is no "way around" an impurity. This is an important, topological, aspect of one-dimensional systems. Another interesting aspect of one-dimensional magnetic systems is the
fact that in some cases, exact solutions for the thermodynamic proper-ties of these systems exist, even for mixed systems. This is especial-ly the case for ld systems which can be described by the "Ising" model or by the "classica! Heisenberg" model. Various Co++ systems belong to the first category, whereas the second model is often
real-. ++
1zed by Mn systems. Therefore we have performed our experiments on these two kinds of magnetic systems mixed with weakly or non-magnetic atoms.
It appears that at low temperatures the thermodynamic próperties of one-dimensional magnetic systems are largely dominated by the develop-ment of strong correlations between the magnetic modevelop-ments in the chain. The development of these correlations depends on the particular character of the magnetic interactions, -e.g. the presence of anisotropy - and may be influenced by an externally applied magnetic field (Boersma et
al.,
1981). It is obvious that in impure one-dimensional magnetic sys-tems these correlations and thus most of the thermodynamic properties, such as the susceptibil.ity or the specific heat, will be strongly affected.Until now we have only considered isolated magnetic chains. In reality, however, these ebains form a part of a three-dimensional crystal and they will be coupled to each. other by a weak interchain interaction. At intermediate temperatures, where the thermal.energy is about equal to the intrachain interaction energy, the thermodynamic properties are hardly affected by these weak interchain interactions. At lower temperatures, however, these interchain .interactions are enhanced by the correlations in the chain and the system will, ultima-tely, change over to a three-dimensional order.ed state. Ther.efore the maasurement of the three~dimensional ordering temperature m~y
indirect-ly yield information about the development of the correlations in the chain.
As already mentioned above, we will start in chapter II with a short review of random magnetic systems, where we will pay special attention to the classification of the various concepts encountered in the literature, and to quasi one-dimensional random magnetic systems•
After a description of the various experimental methods and the preparatien of the mixed crystals in chapter III, we shall present
the theoretical methods used to compute the behaviour of quasi one-dimensional mixed magnetic systems with anisotropic interactions. In chapter V various Mn++ systems, mixed with Cu or Od, will be described and special attention will be paid to the magnetic properties of the quasi one-dimensional system (CH
3)2NH2MnC13• Chapter VI will be devoted to quasi ld Co++ systems, mixed with Cu. In chapter VII we will describe the mixture CsMn(Cl
1_xBrx)3.2H2
o,
of which the 3d-ordering temperature has been measured for 0 <x< 1.Parts of chapter IV, V and VI have already been published by Schouten et a[., (1980); Takeda et a[., (1978) and Schouten et aZ.,
CHAPTER 11
REVIEW OF RANDOM MAGNETIC SYSTEMS
2.1. Introduetion
The experiments reported in this thesis deal with the effect of "impur;ities" in magnetic systems. Depending on the details of the nature of these impurities and the way they are introduced in the system, and depending on the host system to start with, these "impurities" cause some kind of randomness. The resulting effects, especially in low dimensional systems, and the relation with the existence of a long range ordered state farm the main subject of this thesis.
During the last decade, several theories have been developed in which randomness is a manageable physical parameter. Also an in-creasing amount of experiments have been performed to determine the properties of materials with a random nature. An extensive review of "randomly disordered" systems, though largely devoted to experiments on "random alloys" and related systems,as wellas various relevant theoretica! techniques, has been given by Elliott et al. (1974). However, na camparabie review has been presented for random maanetie systems. Mainly for this reasou we thought it worthwhile to start this thesis with a short review of relevant theories and results of recent experiments on random magnetic systems. The secoud reasou is that we will try to contribute to a more consistent nomenclature of the various "impurity" and "random" systems.
The organization of this review is as follows. Insection 2.2. we will try to
field of random magnetic systems by
structure in the large two classes of random magnetic systems: amorphous magnetic systems and regular random mag-netic systems. In that sectien we will further introduce various concepts related to these two classes of random magnetic systems. These concepts will frequently be used tbraughout this chapter and the remaining part of this thesis.
The last two sections of this chapter will be devoted to experi-ments and theories on two- and three-dimensional systems and on quasi
one-dimensional systems (sections 2.3. and 2.4., respectively). The reader should note that in this review the most recent experiments and theories have not been included. For practical reasans we had to limit this review to material published up until early 1980.
2.2. Random magnetic systems
2.2.1. Introduetion
The Hamiltonian which describes a system of interacting spins can generally be written as
H
-I
E(S.,S.)-
I
{s.~.s.
+van
f.s.}.
<ij> l. J { l. l. l. IJ l. l
(2. I)
where <ij> denotes that each pair of spins has to be counted only once. The first term describes the interaction energy between the spins, the second term accounts for the single ion anisotropy energy, and the last term denotes the additional energy resulting from an externally applied magnetic field
H.
Depending on the particular form of the various terms and of the particular site dependenee of the first term, this Hamiltonian describes a non random or a random magnetic system. We will mainly use the first term to distinguish between the different magnetic systems.A non random magnetic system consists of a periodic lattice, where corresponding sites in the various unit cells
kind of magnetic atom. The interaction energy defined function for each pair of spins <ij>
are occupied by one
E(S.;s.)
is a welll. J
and has the same
"periodicity" as the lattice, in contrast to random magnetic systems.
Roughly speaking randomness aan be introdueed in systems
i f at least one of the two basic elements of non random systems viz.
~he of the Zattice or the presenee of only one kind of
magnetic atom at eorresponding is relaxed.
First, a magnetic system displays randomness if it does not have a periadie structure, either with respect to the distribution of the magnetic atoms in space -no periadie lattice- or with respect to the spatial distribution of the magnetic interactions. Secondly, randomness
.
may arise if the magnetic system consists of a mixture of two ormore kinds of magnetic atoms. Of course, one may notice that since both features can be present simultaneously they do not form a
sufficient basis to distinguish the different kinds of random magnetic systems. Therefore we will consider the magnetic interaction in more detail.
In random magnetic systems the interaction energy
E(S.,S.)
will1 J
vary aperiodically and randomly. An additional distinction between the types of random systems given above arises if we take into account the range of the interactions, which can be infinite or restricted to the immediate surroundings of the magnetic atoms. If we combine this range with the presence of regularity in the structure of the magnetic system, we arrive at the following definition of two different classes of random magnetic systems:
The amorphous magnetic systems. These systems do not display any regularity in their magnetic interactions, either because they do not possess a periadie lattice, or because this periodicity is obscured by the long range and the particular functional form of the magnetic interactions.
The reguLar random magnetic systems. These systems consist of a periodic lattice of sites, which may be occupied by various kinds of magnetic or nonmagnetic atoms. The relevant magnetic in-teractions are restricted to nearest neighbour sites only.
We should warn the reader that the difference between these two classes of random magnetic systems will not always be as absolute as perhaps suggested. In fact, we have chosen for a division which makes the subject more manageable, and also facilitates the classification of the investigations reported in this review. We will now continue this section by giving a short treatise of amorphous magnetic systems, followed by a survey of the most relevant concepts related to regular random magnetic systems. The regular random magnetic systems will thereafter be reviewed more extènsively in sections 2.3. and 2.4., where we will consider two- and three-dimensional systems, and guasi one-dimensional systems, respectively.
2.2.2. Amorphous magnetio systems
According to the definition given in the introduction, amorphous magnetic systems do not show any regularity of magnetic interactions, not only with respect to their magnitude and direction but also with respect to their spatial extent. As a consequence of this irregularity the directions of the magnetic moments vary randomly from site to site,even at zero temperature,and there does not exist any long range ordered state. The amorphous magnetic systems encompass very different materials, such as dilute random substitutional alloys, amorphous metallic alloys, amorphous magnetic semiconductors, and conventional glasses. Note that we did not use the term spin glass, which forms the most important concept in the field of amorphous magnetic systems but is also often confused with it.
We will first give several definitions of the term spin glass and next discuss them.
1. The definition given by Andersen (1973), one of the inventors of the term: "A spin glass is a very apt term to describe the entire class of magnetic alloys of moderate dilution in which the magnetic atoms are far enough apart so that the magnetic structure no longer resembles that of the pure metal, but close enough so that their exchange interactions dominate other ener-gies such as the Kondo effect and other free electron inter-actions".
2. A spin glass is a system, which displays experimentally the same particular physical properties as the systems referred to in the definition of Anderson.
3. A spin glass is a system in which the magnetic interactions are random and competitive, e.g. a random mixture of ferro- and antiferromagnetic interactions.
The standard example of a spin glass, in agreement with definition 1., is the metallic Cu-Mn alloy with a molar concentratien of Mn atoms in the range 0.5 - 10%. The most characteristic properties of such a spin glass are the sharp cusp in the susceptibility at the so-called freezing temperature Tc, no evidence either for antiferro- or ferromagnetism (Kouvel, 1961), a small remanence of the magnetization aftera large magnetic field is applied (Tholence and Tournier, 1974), and a linear
(cl M (in ,....j ct H~IO koe
F'ig. 2.1. TI'KI
Temperature variation of the magnetization for various Cu~~ a~~oys
(designated by the at. per aent Mn), measured by Kouve~ (1961) in a magnetia ~d of 10 kOe. 'l'he maximum in the magnetization is due to the aharacteristia cusp in the susaeptibility at the freezing temperature.
specific heat at low temperatures (Zimmerman and Hoare, 1960). In figure 2.1 we reproduced the susceptibility of Cu-Mn alloys, measured by Kouvel (1961).
For some time, the physical model that was generally accepted for these substances was based on a rather peculiar behaviour of the ex-change interactions as function of distance. These interactions were assumed to follow the RKKY-theory (Rudermann-Kittel-Kasuya-Yosida), oscillating with an amplitude falling off as !/r3
, (Fisher, 1977).
The second definition given above arose when the particular spin-glass properties were also found in nonmetallic substances, in which the RKKY interaction cannot exist (Levy and Hasegawa, 1977).
Edwards and Anderson (1975) have shown that it is nat essential that the interactions are of the RKKY-type, but that it is vital that the interactions are of random sign. The fact that the various inter-actioris via different paths between two given spins can compete is the essence of the spin glass phenomena. This also explains why in totally different non-metallic systems spin glass like properties are predicted if they contain a random mixture of ferro- and
netic interactions (Cabib and Mahanti, 1974; Matsubara and Katsura,
1977). These recent theoretica! developments lead to the third definition, which in our is the most appropriate one, as it is based upon the physical nature of interaction mechanism.
2. 2. 3. systems
The regular random systems have a periodic lattice of sites i, which may be occupied by various kinds of magnetic or non-magnatie atoms. The relevant magnet ie interaction
E(S. ,S.)
is restricted~ J
to pairs <ij> located at naarest neighbour sites. The term "regular" indicates that some of the syrnmetry of the regular host lattice is reflected in the spatial pattern of the exchange interactions, due to their short range character. The term does, however, not refer to the magnitude of the interactions. In the remaining part of this reviewwe will restriet ourselves to regular random syste~s for which the magnetic interaction can be written as
+ + +
is.
J ..
s.
i,j are nearest neighboursE(S. ,S .)
~ J ~ ~J J (2.2)
0 otherwise •
The randomness in these systems arises because the exchange tensor
t . +
J .. and the sp~n S. may vary randomly.
~J ~
In this section we will give a survey of the different terms and concepts frequently used in conneetion with regular random
systems:
- the type of.magnetic interaction: Heisenberg, Ising or XY - mixed and dilute systems
- site and bond mixed systems - the quenched and annealed limit.
Strictly speaking, the items listed above are not all of the same im-portance or the same level. Especially item one may be equally well applicable to non random systems. However, we thought it worthwhila to
consider this item, since it is crucial for the work described in this review and in this thesis. The other three items are more or less related to the experimental realization of regular random mag-netic systems, though some of them have theoretica! implications also.
Type of magnetia interaation
The exchange interaction given by Eq. (2.2) can generally be simpli-::i:
fied by assuming the exchange tensor J •• to be diagonal, in other words ~J
we may write .
.... .... E(S. ,S.)
~ J (2.3)
By restricting the number of independent elements Jaa we arrive at the following well-known classification scheme of the different types of magnetic interactions. 0 isotropie Heisenberg isotropie XY Ising (Z). (2.4)
This classification scheme has proven to be very adèquate in the description of magnetic interactions between localized ions in isolating systems. To bring more consistency in the nomenclature for the various magnetic model systems, it is proposed to call the last type of magnetic interaction: Z (Boersma et al., 1981).
If the exchange parameters for the relevant spin components are not exactly equal but differ slightly, we speak of an anisotropic
Heisenberg or an anisotropic XY interaction.
Mixed and dilute systems
Judging from the inconsistent use of the various narnes- "random", "impure", "dilute" (Krzeminski, 1977; Tonegawa, 1976; Thorpe, 1975; Hone et al., 1975) - it seems necessary to specify these concepts more clearly. To this end we define two classes of regular random magnetic systems, which we will use throughout this thesis. We define a mixed magnetic system as a regular random magnetic system composed of two different systems, of which at least one is magnetic. The special case in which a magnetia system is mixed with a nonmagnetia system, is defined as a dilute magnetia system. Note that we defined the dilute magnetic system as a special case
of
the mixed magneticsystems, which is in most theoretica! calculations in fact -obvious. In reality, however, an essential difference exists between both systems. In dilute magnetic systems with nearest neighbour interaction only there appears to be a minimum concentration of magnetic atoms, below which no long range order can occur. Mixed magnetic systems do, in genera!, not have such a critical concentration.
In experimental investigations often the terms "impurity" and "impure system" are used. These somewhat confusing concepts are used, for mixed as well as dilute systems, if the concentration of one of the compounds, the impurity compound, is much smaller than the concen-tration of the other, the host compound.
Site and bond mixed systems
This distinction is a logica! consequence of the two essentially different ways in which two magnetic systems may differ from each other. First, the magnetic interaction between two ions located at corresponding sites in both systems may be different. If such systems, which have different magnetic interactions, i.e. different bonds, are mixed, we obtain a bond mixed syetem. On the other hand, two magnetic systems may be composed of different magnetic ions, viz. two systems A and B having spins SA and SB' respectively. A mixture of such a system is called a site mixed system.
Theoretically, this distinction is useful, since calculations on site and bond mixed systems often yield different results. Experimen-tally, however, the distinction is somewhat academie, since it is hardly conceivable to substitute a second type of magnetic atom without changing the exchange interactions. A mixture of two spins and SB will automatically lead to a mixture of three "bonds", i.e. three exchange parameters JAA' JAB and JBB' The complement, a solely bond mixed system is realized easier, e.g. by replacing the ligand ion in a superexchange p~th.
One should note that the concepts "site" and "bond" appiy to dilute magnetic systems also.
Quenched and anneaZed Zimit
In relation with the distribution of the different atoms or the interaction parameters, it is important to distinguish between the quenched and annealed limit, which are of experimental origin but also have some theoretica! implications. The distinction between these two cases has been first discussed by Brout (1959), and can be summarized as follows.
The free energy of a system which is heated to infinitely high tem-peratures and cooled down infinitely
given by
F
=
-kT<ln<Z> >Q s c
to a temperature T is
(2.5)
where < > denotes a configurational average and < a spin average. c
In this system the different magnetic atoms are frozen randomly at their positions at T
=
oo,
according to a real statistica! distribution. This distribution does not necessarily correspond to the lowest possible value of the free energy at any other temperature and is called the quenched limit.In the annealed limit, the system is allowed to reach the lowest free energy by varying the temperature and other relevant parameters infinitely slowly. This should also include a free movement of the magnetic atoms through the crystal. The free energy is then given by
kT ln<Z>
s,c (2.6)
The importance of this distinction can be illustrated by the following example. In a mixture of liquid 3He and 4
He below a eertaio tempera-ture, the free energy is lowered by a condensation of the 3He atoms in
the so-called concentrated phase. One can imagine that in an analogous way in a mixture of two different magnetic systems a condensation may occur in two phases, each containing one kind of magnetic atoms.
In crystals a free movement of ions is never possible. Hence one may assume that, if the temperature at which the crystals are grown is sufficiently high compared to the magnetic interaction energy, the mag-netic behaviour will correspond to the quenched limit.
2.3. Two- and three-dimensional systems
2.3.1. Introduetion
In the preceeding section we divided the random magnetic systems in two different classes: the amorphous magnetic systems and the regular random magnetic systems. As already mentioned in that section we will further restriet this review to the regular random systems. At this point we like to emphasize again that the differences between the two classes will not always be quite clear or distinct, and the reader will be confronted with some examples of, for instance, mixed magnetic systems, which display spin glass-like properties under certain cir-cumstances.
The division between two- or three-dimensional (2d or 3d) systems and one-dimensional (!d) systems logically results from the fact that the first two systems may display long range magnetic order, while the third never does. Also topologically there are essential differences.
Most studies on 2d and 3d regular random magnetic systems are devoted to site dilute systems, i.e. regular magnetic host systems mixed with non-magnetic atoms. The main topic of these studies con-.cerns the existence of a long range ordered stat~ at low temperatures, especially the location of the ordering temperature as a function of the degree of dilution. In this context the concept "percolation" wil! appear to be important. In recent years also the influence of an additional externally applied magnetic field has been taken into ac-count.
With respect to mixed magnetic systems, most attention has been given to systems with competing interactions, in relation with the occurrence of new phases, which in some cases leads to rather exotic phase diagrams and the already mentioned glass properties.
The organization of this sectien is as follows. We start with a review of theories on dilute systems, of which we will mainly treat the p-T phase diagram (concentration of magnetic atoms pt versus
tThe notatien p
=
concentratien of magnetic atoms is most frequently used in theoretica! calculations. In experimental investigations generally the notatien x concentratien of nonmagnetic atoms, x=
1-p, is used. In this review both notations will be used, depending on the context.ordering temperature), and the influence of dilution on the magnetic H-T phase diagram. This review will be followed by a survey of the experimental verifications. The last part of the section, 2.3.3., will be devoted to the experiments and theories on mixed magnetic systems.
2.3.2. Dilute magnetic systems
Over more than 20 years dilute magnetic systems have been studied, especially the influence of dilution on the ordering temperature of "magnetic crystals" (Behringer, 1957). The first theories, using a simple molecular field approximation, yielded a linear relation between the ordering temperature Tc and the concentration of magnetic
atoms p, for all values of p. However, using simple common sense ar-guments, one can easily show that as long as we are dealing with short range nearest neighbour interactions only, the onset of magnetic ordering .cannot occur at zero concentration. A certain minimum amount of magnetic atoms has to be present. Mathematically this is called a percolatien problem. Dilute magnetic systems farm an important class of physical systems which are very suitable to study percolatien processes experimentally.
Though percolatien processes farm a pure topological problem, we will start this subsectien with a short discussion of the relevant concepts of percolation theor~ because they will be frequently used in the remaining part of this section. Next we will treat the p-T
phase diagram. This diagram can theoretically be approximated in two complementary ways: the Zow concentration approach and the high con-centration approach. The last part of this subsection will be devoted to the influence of dilution on the H-T phase diagram, followed by a survey of some relevant experiments, which have been performed up till now.
We will omit a discussion of theories dealing with the oritioaZ
properties of dilute magnetic systems. The interested reader is
re-ferred to articles of among others Harris (1974), Reeve and Betts (1975), Stauffer (1975), and Lubensky (1977).
PercoLation
Without going into the details of the mathematica! formulation of the percolatien problem, we will discuss some of its relevant concepts.
Consicier a lattice in which some of the sites are occupied by particles, the remaining sites being vacant, see figure 2.2. The occu-pied sites fall into a number of clusters, two such sites belonging to the same cluster if there is at least one chain of occupied sites connecting them. The size of the clusters will depend on the fraction p of sites being occupied. Intuitively one expects the mean size to increase with p and for a finite crystal the maximum cluster size will be reached when p
=
1. In case of a crystal which extends infinitely in all directions, the situation is somewhat different, as the cluster size is no langer limited. The critica! concentratienas the largest value of p for which all clusters have with probability 1, a finite size. For p >pc there is a non-zero proba-bility that a given occupied site belengs to an infinite cluster.
In dilute magnetic systems, the occupied sites have a magnetic moment and the "vacancies" are occupied by nonmagnetic atoms. As long as all magnetic atoms are divided into finite clusters, no long range ordered state can arise,even at zero temperature, since the nearest neighbour exchange interactions couple only magnetic atoms
to the same cluster. The critica! concentration, also called the limit, farms a lower bound for the concentratien of magne-tic atoms at which long range order can occur. Notice, this condition is necessary but not sufficient for the existence of a long range ordered state, as we will discuss below. The critica! concentratien can be calculated exactly only for a restricted number of lattices (mostly bond dilute systems).
An important parameter which determines pc is the number of nearest neighbours z. This is illustrated for a simple finite two-dimensional lattice in
sites for p
2.2. We generated a random configuration of occupied 0.54. Case A corresponds to a simple square lattice with z
=
4, for which p~=
0.58 (Essam, 1972). Inspeetion of this figureshows that several small clusters have been formed, according to p < pc. Case B shows the same configuration on a triangular lattice with z 6, for which pc 0.50 (Essam, 1972). The number of clusters
Pc =0.58 p.o.5~ z-4
Fig. 2.2.a.
A particular -randomly
generated-Pc • 0.5 P.O.St. z-6
Fig. 2.2.b.
The same on a
tri-lattice (z
=
6) for whicha square lattice (z 4) illustra- p >pc' the presence of
presence clusters with an a luster.
size if p <
lattice, illustrating that p > Pc·
Another important quantity for the description of the physical properties of a dilute system is the percolatien probability P(p), defined as the probability that a given occupied site belongs to an infinite cluster. One can easily deduce that P(p) 0 for p < pc and P(p)
=
l for p=
1. In dilute magnetic Ising systems P(p) appears to be related to the spontaneous magnetization at zero temperature (Elliott et al., 1960).A survey of the percolatien theory and the relation to dilute systems is given by Essam (1972, 1980). An excellent review of percolatien processes in many other branches of
by Frisch and Hammersley (1963).
is presented
Before proceeding we remind the reader that percolatien theory only prediets the concentratien range, dependent on the type of lattice, where long range magnetic order can occur. This is a purely topclogi-cal problem, viz., an infinite cluster of magnetic atoms is presentor not. In real magnetic systems the particular kind of magnetic inter-action and - of course - the temperature determine whether such an
ordering will occur. In the literature it is generally assumed that the concentratien at which T
c 0 coincides with the percolatien limit
(Elliott et al., 1960), but this cannot be proven from first princi-ples (Rushbrooke et al., 1972). As already mentioned above, percolatien
theory just gives a lower bound for the concentratien at which an in-finite cluster of "coupled" magnetic atoms is present.
The p-T phase diagram
The experimental and theoretical studies on dilute magnetic systems can be explained most clearly in conneetion with the p-T phase diagram. In figure 2.3 a typical predietien for the behaviour of the ordering temperature as a function of the concentratien of magnetic atoms p is plotted for three different systems. Curves marked 3d and 2d denote a fictitious three- and two-dimensional magnetic system, respectively, which display long range order, see, for instance, Elliott and Heap
(1962). The third curve will be discussed separately.
The difference between the curves "3d" and "2d" arises mainly from the difference of the critical concentratien below which no magnetic long range order occurs. This clearly demonstrates the relation between the number of nearest neighbours and the critical concentration, in agreement with the percolatien theory mentioned above. The percolatien theory prediets a lower pc if the number of nearest neighbours z in-creases (z will be generally higher in three- than in two-dimensional systems). Note we plotted here two fictitious systems, the exact value of pc varies for each particular lattice (fee, bcc, sq, etc.; see table 2.1).
lattice Pc z
sq square 0.58 4
t triangular 0.50 6
SC simple cubic 0.31 6
bcc body centered cubic
I
0.24 8 fee face centered cubic 0.20 12Table 2.1. Critical concentration pc, i.e. the percolation limit, for several lattices with different number of nearest neigh-bours z. (Essam, 19 72).
Fig. 2. 3. p
Conjectured p-T phase diagram for three different dilute magnetic systems with nearest neighbour interaction, which display long range order:
a three-dimensional (3d) system; a two-dimensional {2d) system; a quasi one-dimensional (ld) system.
At higher temperatures (above the dashed line the latter system dis-plays ld eharacteristics, approximating I :::: co • At lOJ"Jer temperatures
g
(below the dashed line) where T is of the order of the interchain in-teraction energy, the system behaves as a 3d system.
The particular form of the T (p) curve depends on the kind of
c
magnetic interaction and is the subject of the various theoretica! and experimental studies, which will be discussed in the remaining part of this subsection.
Now we return to the third curve in figure 2.3, which denotes the beha~iour of T (p) for a quàsi one-dimensional magnetic system. This
c
is a real magnetic system of which the magnetic properties are largely determined by the strong exchange interaction J in one crystallographic direction, whereas in the other directions the exchange interactions are much weaker.
In real quasi ld systems, the 3d ordering is a consequence of the
presence of the weak interchain interacticns. When the temperature is of the order of the intrachain interaction J/k, strong correlations
between the spins in the same chain will develop. These correlations increase at lower temperatures, while the interchain correlations are still very small. Simultaneously the interchain interactions will be enhanced by these highly correlated chains. This proces will lead to a 3d ordered state at a temperature which is much higher than the interchain interaction J' /k.
When we start to dilute such a system the effect of the nonmagnetic atoms will be primarily a reduction of the long correlation in the chain, - a topological, ld behaviour and thus a reduction of the 3d ordering temperature. This effect dominates in the high concentratien region of the p-T diagram indicated by the dashed line. The more the sytem can be described by a I d sys tem (J' /J << I) the larger the value of
T
~p)
.(c
since for real Jd systems the value of
dilution, the correlation in the chain
is infinite. With increasing become of the same order as the correlations between spins from different chains, the inter-chain correlations. The typical ld behaviour will now disappear and the system will behave as a strongly diluted 3d system, with strong and weak interactions. The behaviour in the p-T phase diagram will be analogous to that of other 3d systems, but at a considerably lower temperature than the 3d ordering temperature of the pure·system. In figure 2.3 this is the region below the dasbed line. We will reeurn to the p-T phase diagram, especially the high concentratien region, more extensively .in chapter IV.
There are two important aspects of the p-T phase diagram, which are, in genera!, calculated in two different ways; these are the critica! concentratien and the slope I • The first can be most accurately
g
calculated from the low concentration approach, while the latter can
be better calculated from a high concentration approach.
Low concentration approach
In order to calculate the critica! concentration of a dilute mag-netic system, one usually expands the susceptibility at finite tem-peratures in powers of the concentration. Assuming that magnetic
long range ordering will occur, one looks for a divergence, which would indicate the onset of such an ordering. In this way pc has been calculated for several 3d dilute Heisenberg and Ising systems by Elliott et al., (1960), Morgan and Rushbrooke (1961), Elliott and Heap
(1962), Morgan and Rushbrooke (1963), and laterfora 3d di'lute XY
system by Reeve and Betts (1975). All the calculations except one (Elliott and Heap, 1962) are restricted to ferromagnetic interactions only. Without going into details, we will summarize some of the basic assumptions underlying this approach.
Consider a very dilute magnetic system, consisting of N sites, with a molar concentratien of magnetic atoms p. These atoms will be present in clusters of various magnitudes and geometrical configurations. A cluster is defined as a set of n magnetic atoms, arranged in such a way that each magnetic atom is conneeeed to at least one other atom of that cluster by a nearest neighbour exchange bond, but none of them is conneeeed in any way to the other (pN n) magnetic atoms. The suscep-tibility is now evaluated as a sum of the susceptibilities of the in-dividual clusters
x<T) (2.7)
where Ana denotes the number of clusters with the geometrical confi-guration a, consisting of n atoms, and Xna(T) represents the
suscepti-bility of this cluster. It can be shown that Ana is a polynomial in the concentratien p and that x(T) can also be written as a polynomial
x(T) '\ L ai(T)p i i
(2.8)
The onset of magnetic ordering is investigated by determining the singularities in this expression for a given temperature T and concen~ tration p.
The evalu~tion of ~a(T) is restricted to clusters with n < 5 for 3d systems - Heisenberg as well as Ising - and to n < 9 for 2d sys-tems.·Therefore the uncertainty in pc varies from JO- 20 %. Within this range the value of pc appeared to be the same for Ising and Heisenberg systems, a fact which has been proven by Elliott et al.
(1960).
The approach described above is based on the assumption that a di-vergence of X(T) indicates the transition to 3d long range order, and the assumption that also in dilure systems X(T) will diverge at the 3d ordering temperature. A large number of experiments have suggested that various thermodynamic properties of pure systems diverge at the critica! , i.e., the point at which a transition to long range order occurs. This evidence is corroborated by the apparent consisten-cy of various numerical calculations and renormalization group theory.
A divergence of the susceptibility, however, does - on itself not necessari imply the onset of long range order, as ~s obvious from
the results for the 2d Heisenberg model, which does not order
(Mermin and Wagner, 1966), although a divergence of the susceptibility is prediered (Elliott and Heap, 1962). A similar situation holds for the 2d XY model (Stanley and Kaplan, 1966).
With respect to the secend assumption, we like to note that dilute magnetic systems, which have randomly varying microscopie properties, might fall into a different category, since it is that the cri-tical behaviour is much more complicated. There is in fact no direct evidence that a simple critica! point exists in all cases; there may be some extended temperature region over which the transition is smeared out. The disappearance of a singularity is in fact shown by McCoy and Wu (1968) for a simplified bond mixed two-dimensional system. The singularity in the specific heat is smoothed into a funct-ion which is infinitely differentiable at Tc' though not analytic. The HTE method is sometimes used to calculate Xna(T) in Eq. (2. 7), (Reeve,
1976; Rapaport, 1972; Morgan and Rushbrooke, 1961).
High aoncentration approach
In the theoretica! calculations related to the high concentratien side of the p-T phase diagram, i.e. at p
system as a perturbation of the pure p
I, one regards the dilute state applying a modified molecular field approximation (McGurn, 1979; Tahir-Kheli and McGurn,
1978; Yeomans and Stinchcombe, 1978; Harris, 1974). The methods should only be used for relatively high concentrations of atoms,
(1-p) << 1. In the introduction, we referred to simple molecular field approach which prediered a linear.behaviour of Tc(p) with p: p.Tc(1).
latt. interaction Ig methad reference
sq Ising 1.329 HTE Harris, 1974
OJ
...,
sq Ising 1 .343 other Nishimori, 1979
;l ...
....
Ising 1.35 GF McGurn, 1979 "d sq "d-
--
- -
- -
1 - - --
--
- -
-~ SC Ising 1.20 GF McGurn, 1979 0 ..0 SC Ising 1.06 HTE Harris, 1974
sq Ising 1.64 other Stoll et al., 1976
sq Ising 1.387 GF Yeomans et al., 1978
sq Ising 1.33 GF McGurn, 1979
sq Ising 1.20 cluster I dogaki et al., 1978
sq anisotropic Heisenberg 3.142 GF McGurn, 1979
- -
- -
-
- - - -
-
I - -- -
--
-se Ising 1.00 cluster Morgan et al., 19,61' 1963
SC Ising 1.00 cluster Elliott et al. • 1962
se Ising 1.00 cluster Rushbrooke et al., 1961
OJ
Ising 1.05 cluster Idogaki et al .• 1~178
..., SC
;l
...
XY 1.22 HTE Reeve et al, 1975
....
SC"d
<ll SC Heisenberg 1.30 cluster Morgan et al., 1961. 1963
...,
....
Heisenberg 1.00
Ul SC cluster Elliott et al.' 1962
SC Heisenberg >I cluster Rushbrooke et al. • 1961
SC anisotropic Heisenberg 1.68 GF McGurn, 1979
-
- -
-
- -
--
-
f - -- - - -
-bcc XY I. 15 HTE Reeve et al., 1975
bcc anisotropic Heisenberg 1.51 HTE McGurn, 1979
- -
- -
-
- -
- -
-
I - --fee Ising 1.02 HTE Rapaport, 1972
fee XY 1.164 HTE Reeve et al., 1975
fee Heisenberg 1.0 MF Behringer, 1957
TabZe 2.2. TheoretiaaZ values of the slope of T0(p) at p 1, I g for
?
various two- and three-dimensional dilute magnetia systems. The systems are arranged aaaording to increasing number of neares t neighbours z (af. tab le 2. 1).
(HTE =high temperature expansion~ GF Green's funation,
The authors mentioned above use newly developed methods such as Green's function theories and renormalization group methods. Their results, though restricted to the slope of Tc(p)/Tc(l) at p =I, Ig, agree with each other for corresponding cases. The agreement with the predictions obtained from the low concentration approach, cf. Eq. (2.8), is slightly worse.
In table 2.2 we summarized the values of I , predicted by several
g
approximations, for various two- and three-dimensional dilute magnetic systems. Inspeetion of this table shows that the differenc theoretica! techniques do not always yield the same results. There is, however, a significant difference between two- and three-dimensional systems.
I is somewhat larger for two-dimensional systems. This may be
relat-g
ed to the generally higher critical concentration. A remarkable high
value of I is only predicted by McGurn (1979) for the anisotropic
g
Heisenberg system, especially in two dimensions. We will return to this point in the experimental part of this section. The table clear-ly indicates the need for experimental verification. Same relevant experimental results will be reviewed in section 2.3.3.
The H-T phase diagram
During the last years, there is an increasing interest in the mag-netic phase diagram of antiferromagmag-netic systems, especially in relation to the thermadynamie behaviour in the neighbourhood of the critical lines or points. In figure 2.4 we plotted the phase diagram of a simple three-dimensional weakly anisatrapie antiferromagnet, with a magnetic field applied parallel to the easy axis. In this phase diagram three different regions can he distinguished, separated by three so-called phase boundaries: the antiferromagnetically ordered phase (AF), the spin flop phase (SF) and the paramagnetic phase (P). The phase boundaries meet in the bicritical point.
The effect of dilution on the AF-SF transition at T
=
0 iscalcula-ted by Kaneyoshi (1975) using a spin wave analysis. He prediets that HSF strongly decreasas with increasing dilution. The influence of di-lution on the entire phase diagram is calculated within a Green's
paramagnet ie spinflop I b1critical point
i
!
antiferromagnet ie 2.4. 0- r
Schematic H-T diagram of a anisatrapie antiferromagnet.
The field is applied parallel to the easy axis.
Tahir-Kheli et al. (1976). They predict that the AF-Pand the SF-P
transitions depend much stronger on dilution than the AF-SF transition. In chapter V we will show that the latter conjecture is nat confirmed by our experiments on quasi one-dimensional systems.
E:x:perimental resuZts
In table 2.3 we summarized various compounds in which the influence of dilution has been studied. In the same table we indicated the type of interaction (Heisenberg (H), Ising (I) ar XY), the kind of lattice, and the reported values for bath the critical concentration p~ and the slope I •
g
The number of compounds is relatively small. This is very likely due to the fact that in general it is very difficult to dilute magnetic systems with arbitrary fractions of nonmagnetic atoms. Moreover, in cases the magnetic properties of the host systern are changed, due to changes in the local crystal field parameters in the neighbourhood of an impurity. This fact on itself has been the subject of studies on dilute systems (Ikeda and Shirane, 1979 on Rb
2Co Mg1 F4).
. p -p
latt. compound pc I exp. g sq Ni(NH 3)2Ni(CN)4·2C6H6+Cd I 0.65 1.2
x.c
Nagata, 1974 sq K 2CoF4+Mg I 0.57 1.6x
Breed, 1970 sq Rb 2CoF4+Mg I < 0. 7 1.95 C,ND Suzuki, 1978 sq Rb 2CoF4+Mg I-
ND Ikeda, 1979 sq Mn(NH 3)2Ni(CN)4·2C6H6+Cd H 0.69 1.01 x,c Kitaguchi,1978 sq Mn(HC00) 2·2H20+Mg H 0.59 1.0 c Takeda, 1970a sq K 2cuFe4+Zn H <0.78-
ND Wagner, 1978 K 2MnF4+Mg H 0.59 3 X,NMR Breed, 1970 K 2MnF4+Mg H-
-
ESR Yokozawa, 1978 Rb 2MnF4+Mg H <0.54-
ND Cowley, 1977 Birgenau, 1976 Ni(OH) 2+Mg H 0.1~
x Enoki, 1975,1978 SC CoCs 3ct5+Zn I <0.35 1.1 c Lagendijk, 1972 SC Co(C 5H5No)6(Cl04)2+Zn XY <0.55 1.3 c Algra, 1976,1979 SC KMnF 3+Mg H 0.31 1.3 x,NMR Breed, 1973 bcc CoF 2+Zn I <0.5 1.3 NMR Baker, 1962 bcc MnF 2+Zn H <0.25 1.3 NMR Baker, 1961 bcc FeF 2+Zn H 1.0x
Wertheim, 1966 fee MnS lH 0.13 1.4 x Heikens, 1976Table 2.3. Experimental values of the critical coneentration the
s of T (p) at p = 1~ I , for various quasi 2d and Jd
c g
eompounds. The type of interaction has been indicated by I (Ising) > H _(Heisenberg) or XY.
(C speeifie heat~ ND neutron diffraetion).
Note that in the the first author is given only.
We will now discuss the experimental results in relation to the theoretica! predictions, summarized in table 2.2. The experimental values for in 3d systems only display small variations around the value I. I, In view of the experimental uncertainties, this value cor-responds rather well to the theoretica! predietien given in table 2.2. Note that anisotropy in the interaction has little or no influence in 3d systems, in contrast to systems having a lower dimensionality
as we will discuss below. For Zd systems, I LS significantly larger g
than l, but the experimental va lues do not allow one to decide which theoretica! predictions are correct. Apart from this, one should note that actually these compounds are quasi two-dimensional, and the degree of two-dimensionality may vary considerably among the various com-pounds. Remarkable is the large value I
=
3, reported by Breed et al.g
(1970) for K
2MnpMg1_PF4• This is a reasonably good quasi Zd Heisenberg antiferromagnetic system with a small amount of spin anisotropy (Ikeda and Hirawaka, 1974). The observed value of I agrees with the
predic-g
tion for a purely two-dimensional dilute magnetic system with anise-tropie Heisenberg interaction given by McGurn, (1979\ l t is not clear whether this agreement is coincidental, since no systematic experiments on this type of systems have yet been performed. The remarkable effect of anisotropy, which is hardly present in 3d systems - no significant differences in 1 between Heisenberg and Ising interactions - seems to
g
increase in lower dimensional systems.
The experimental values for the critical concentratien can be com-pared with table 2. l, where we have tabulated the most accurate theo-retica! values which are available (Essam, 1972). It is obvious that the experimental values do not allow more than a qualitative confirma-tion, and the experimental uncertainties preclude any preferenee for one of the theories mentioned above, both for 2d and 3d systems.
2.3.3. Mixed magnetia systems
A mixed magnetic system has been defined in section 2.2 as a regular random magnetic system consisting of a mixture of two different magnetic systems. This may result in either a site mixed system or a.bond mixed system. The distinction between these two types of mixed systems has already been considered in section 2.2.3. The special case of a dilute system has been treated in section 2.3.2.
In this section we will try to give a survey of the many essentially different mixed magnetic systems. One may consider, for instance, mixtures of Heisenberg and Ising interactions, mixtures of different spin values, mixtures of ferro- and antiferromagnetic·interactions, or even combinations of these mixtures. To make the subject more manageable,
one can distinguish two classes of mixed magnetic systems, which will appear to display essentially different properties:
I. Mixtures of two magnetic systems belonging to the same class of model systems, e.g. Heisenberg or or XY, (cf section 2.2.3), in which corresponding interactions have the same sign, and in which a single ion anisotropy, if present, favours the same preferred direction of spin alignment.
2. Mixtures of two magnetic systems in which either corresponding interactions have a different type of anisotropy (e.g. Heisenberg and Ising) and/or a different sign, or in which single ion anisotropy favours different preferred directions of spin alignment.
In contrast to the situation in the first class of mixed magnetic systems, in the second a competition between the various magnetic interactions occurs, giving rise to essentially different behaviour, as we will demonstrate below. Formally,· it is hard to conceive that a competition would occur between pure Heisenberg and an other type of interaction. In all real compounds, however, a certain amount
of dipolar anisotropy is always present, inducing a preferred direction of spin alignment.
Apart from the magnetic interaction between various atoms, the Hamiltonian given by Eq. (2.1) also contains a single ion anisotropy
-+=*+ . . . . .
term SiD Si' One can eas1ly 1mag1ne that there may ex1st m1xed magne-tic systems consisting of magnemagne-tic atoms with different single ion anisotropies. Therefore we included the
defiriitions of the two classes of mixed
ion anisotropy in the systems given above. The theoretica! and experimental studies on mixed magnetic systems have been devoted mainly to the concentratien vs. temperature phase diagram and the occurrence of various types of magnetic ordering. In figure 2.5 two theoretica! phase diagrams, each typical for one of the classes of mixed magnetic systems defined above, are reproduced. Figure 2.5a represents the phase diagram of a site mixed system with only ferromagnetic interactions (Dvey-Aharon, 1980). This kind of mixed system, belonging to category J, displays a continuous evolution
from system A into system B with increasing amount of B atoms (x). The actual pathof T (x) in the x-T-diagram depends only on the relative
c
magnitude of the magnetic interactions JAA' JAB and JBB' The dasbed curves in figure 2.Sa represent calculations by Foo and Wu (1972) on the same subject.
a ?00 1'50 !2 ..::' 100 ~ ..::' Û'50 0 0·50 x 2.5. b A component transition
~
. tetracnt,ca pointJ
100 A p ord_fring ofs
8 1 Ba. The theoretica! x-T phase diagram a mixed Heisenberg system
A Bx -x 1 with ferromagnetic interactions onZy. This system belongs
to the category defined in the text. The along
the vertical axis are scaled to Tc(O), the critica! temperature
of the pure B compound. In aZZ curves = and varies
from 0.1 to 2.5 JBB' The curves represent the prediction by
ifvey Aharon (1980) (Green's technique); the dashed curves
correspond to two different molecular type approximations by
Foo and Wu (1972).
b. The theoretical phase diagraJTI a mixed magnetic system beZonging
from Aharony a dif-to the second category defined in the text,
and Fishman ( 19 76). The A and the B
ferent single ion anisotropy, with D > 0 A B.
Three regions at the Zeft, p small, ordering of the A
pound occurs, and on the right, p large, ordering of the B com-pound occurs. The character of the "mixed ordering"
clear.
is not
Figure 2.Sb shows a phase diagram calculated by Aharony and Fishman (1976) for a mixed magnetic systern belonging to the secend category. The pure systems A and B have a single ion anisotropy which is different in such a way that the easy axes in both systems are mutually perpen-dicular. Due to the "competition" between these two systems three different regions arise in the phase diagram. In the two regions with a
high concentratien of one of the component systerns, A or B, ordering of respectively the A component or the B component is observed. In the intermediate concentratien region a mixed phase may arise. Such a mixed phase is called an "intermediate phase", an "oblique phase" or a "spin glass phase", depending on the particular properties that are observed or expected.
Insection 2.2.2., dealing with amorphous magnetic systems, we de-fined a spin glass as a system in which the magnetic interactions are random and competitive. This is based on a recent theory, which states that spin glass-like properties are caused by such competition (Edwards and Anderson, 1975). This may explain why a spin glass phase in mixed
systems is expected or even predicted though it is, as far as we know, not yet shown experimentally.
Another interesting aspect of the phase diagram of the secoud cate-gory of mixed magnetic systems is the existence of multicritical points, i.e. common points of higher order phase transition lines. (Aharony and Fishman, 1976). This creates new experimental possibilities to check the recently developed theories on critical phenomena.
Various representative theoretica! and experimental studies on mixed magnetic systems reported in the literature are summarized in table 2.4. We have made a subdivision according to our definition of
the two distinct of mixed magnetic systems given above. We
Fig. 2.6.
The experimental x-T phase diagram of KzMnl-xFexF4 measured by Bevaart et al.
( 19 77). K
gMnF
4 is a quasi 2d Heisenberg system ~ith a ~eak dipolar anisotropy~hich causes the spins to be aligned parallel to the c-axis. In K2FeF4 a planar anisotropy, caused by crystal
field is present, forcing the
S // c-axis .02 INTEJtttEOtATE .04 .06 .08 S in ab·plane )( .10 .12 .96 .98 too ol-~1o:---:2~o~-=30~--=40~-so;•6~o T!Kl
spins parallel to the magnetic belongs to the second category
(ab plane). This diagram
