ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS

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ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS

W. Wolf

To cite this version:

W. Wolf. ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-26-C1-33. �10.1051/jphyscol:1971106�. �jpa-00213908�

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ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS

(") W. P. WOLF

Becton Center, Yale University New Haven, Connecticut 06520, U. S. A.

RBsum6. - Les conditions dans lesquelles les intkractions effectives spin-spin peuvent &tre fortement anisotropes sont revues dans le contexte de mesures recentes. On dkmontre qu'une anisotropie peut venir de divers effets d'orbite ou de restrictions imposkes par certains types d'ktats d'ions isoles.

Le cas le plus genkral est trks complexe, mais d'importantes simplifications sont possibles dans des cas particuliers oh les interactions peuvent dtre caractkrisbes par quelques parambtres mesurables.

Abstract. - The conditions under which effective spin-spin interactions may be highly anisotropic are reviewed in the light of recent experimental results. It is shown that anisotropy may result from various orbital effects and also from restrictions imposed by certain types of single ion states. The most general situation is shown to be very complex but drastic simplifications are possible in some cases. For these, the interactions may be characterized by a few parameters which can be determined empirically.

I. Introduction. - The nature of the interactions between magnetic spins has been the subject of inten- sive study for more than 40 years. Until recently it was generally assumed that most systems could be described quite accurately by a Hamiltonian of the form :

with J 9 d, ^%-AaP 9 j, where S, and S, are the total ionic spins. In some cases d and some components of A might be forbidden by symmetry, but in all cases J was taken as the dominant term. The main theoretical discussion revolved around the values of the ^<(exchange parameters ⁾⁾J [I], and an explana- tion of the other terms by perturbation calculations [2]

[3]. Thus d was found to be of order (Ag/g) J a n d A

-

(Aglg)' J , where Ag/g is typically of order lo-'- lo-'. Throughout much of this time it was clear to several authors [3-121 that the basic assumptions underlying this picture did not necessarily apply to all physical systems, but there was a general reluc- tance to pursue possible complications because there was little clear-cut experimental evidence that any extension of the simple situation was really required.

One reason for this lies in the fact that just those systems in which we might expect more complex and anisotropic interactions are frequently complicated by competing effects from crystal fields and spin orbit coupling, so that a critical analysis is very difficult. Thus for example, in some of the early work on rare earth iron garnets 1131 a good fit to the available data could be obtained by taking account of the nine crystal field parameters and only one (isotropic) exchange parameter, whereas it is now clear [12]

that as many as ten parameters may be required to describe that rare earth-iron exchange (see Sec. V below). Another factor which complicates anisotropic systems is that they often tend to have rather weak exchange interactions so that other mechanisms become relatively more important.

It is therefore necessary to establish general criteria for the conditions under which extensions of eq. (1) are essential for explaining detailed observations, and to formulate sufficiently general expressions to describe the interactions in such cases. By applying these to specific systems one may then hope to find situations which can again be described in simple terms, even though the form of the interactions will generally be very different from the usual Heisenberg form.

An appeal in this direction was made by Elliott and Thorpe [14] at the last International Conference and they summarized some of the general possibilities which could result from orbital exchange effects.

At that time they gave little experimental evidence for their rather general statements and it is the purpose of the present paper to relate some of these to real physical examples which have been studied in recent years. We shall also review some other interaction effects not discussed explicitly by Elliott and Thorpe and we shall thus arrive at a set of general Hamilto- nians which can be applied to different well-defined cases.

I t is to be stressed that the role of theory is here to define the form and number of the allowed interaction terms and that it is generally quite hopeless to expect accurate quantitative predictions for the parameters which set the energy scale for the interactions. This is no different from the usual case of isotropic exchange where even the sign can often not be predicted [15, 161 and where even in the most favorable cases uncer- tainties of 100

%

occur [I]. It is thus more than ever necessary to use experimental information to determine the interactions, and in Sec. VI we shall review briefly the 15 or so methods which have been used for highly anisotropic systems. Because the interactions are generally characterized by several essentially independent parameters it is usually necessary to combine the results of a number of different measurements [17, 181 or else look for systematic trends across a series of related compounds [19], but care must be taken in interpreting such comparisons.

(*) Supported in part by the U. S. Atomic Energy cornis- 11. Classification of interacting systems.

-

The form

sion. of the appropriate interaction Hamiltonian depends

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971106

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ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS C 1 - 2 7 on two essentially independent factors : the nature

of the interacting systems in the absence of the interactions, and the physical nature of the mechanisms responsible for the interactions. The total Hamilto- nian describing two interacting systems I and 2 can generally be written as

where 32:') and

xi0)

⁹XI, and the terms

xi1)

^and

xi1)

may or may not be comparable with XI,. The classification of different cases depends on the eigen- functions of X(O) which provide a basis for the des- cription of X12 and X(').

Since $3') is by definition large, we generally consider only one state (or group of states) at a time, and these may be characterized as one of several types :

1) TYPE S. - S-state with negligible orbital admixtures. Total spin S a good quantum number.

2) TYPE Q. - Orbital angular momentum quen- ched. Similar to S-state but other orbital states generally closer, allowing larger admixtures by XI, or Xfl).

3) TYPE L'. - Orbital degeneracy (or near degeneracy) with 2 L' -I- 1 states ; spin orbit coupling weak.

For a free ion L' = L, the total orbital angular momen- tum, but in a strong, highly symmetrical crystal field, states with 0 < L' < L may be found [14].

4) TYPE J. - Spin orbit coupling strong ; crystal field weak. Total angular momentum J = L

+

S a good quantum number.

5) TYPE s'. - Degenerate (or near degenerate) group of states resulting from an interplay of spin orbit coupling and crystal field. In general the states are combinations of several different spin and orbital states but a wide range of situations exists. For rare earth ion the states are mainly admixtures of different IJ,

>

derived from a given J ; for 3 d, 4 d and 5 d ions they may be admixtures based either on type Q or type L' orbital states.

6) TYPE M. - It sometimes happens that some of the ions in a system become strongly polarized by other neighbors or applied fields, so that they tend to a singlet state which corresponds to a magnetic moment in a particular direction

%.

Examples of this are the Fe3+ ions in the iron garnets at low tempera- tures (whose interaction with the rare earth ions will be of interest).

111. Physical interaction mechanisms. - In this section we shall describe briefly the different mechanisms which have been found to be important in one or more real systems and we shall give rough estimates of the orders of magnitude of the corresponding ener- gies. Since the explicit form of the different Hamilto- nians depends on the particular single ion states we shall defer this discussion to the next section.

1) MAGNETIC DIPOLE-DIPOLE COUPLING (MDD). - This requires no discussion, except to remark that this interaction is sometimes rather more important than is often thought. For example in a concen- trated material with high magnetic moments, such as

Tb(OH),, the dipole coupling contributes an energy of 8.0 cm-' to the splitting of the ground state [20].

2) ELECTRIC MULTIPOLE INTERACTIONS (EMI). - The coupling produced by aspherical charge distri- butions in solids is quite complex [21, 241, but there is good experimental evidence that at least the lowest order term, the electric quadrupole coupling (EQQ), may be important. Thus in CeCl,, EQQ contributes about

+

0.1 cm-I per nearest neighbor pair to the energy [25], but in certain other systems (especially those without Kramers degeneracy), the effect could be much larger (See below).

3) VIRTUAL PHONON EXCHANGE (VPE). - This mechanism is closely related to EMI. It differs in that the interacting multipole moments are both induced by distortions resulting from phonons. This mechanism was generally believed to be very weak [22, 26, 281, but Allen has recently shown 1291 that the effect can be very much larger in non Kramers systems with orbital degeneracy (type L' or J), and he has been able to account for the antiferromagnetism of UO, (TN = 30 OK) in terms of this mechanism.

4) EXCHANGE INTERACTION (EI). - This has been discussed in great detail [l] and we only want to draw attention to the well-known and obvious fact [4-121 that exchange interaction will generally depend stron- gly on the particular orbitals in which the individual electrons happen to be. Previous simplifications neglec- ting this dependence merefy resulted from the assump- tion that only one orbital states was involved, so that the individual orbital exchange operators could be replaced by their expectation values in that state.

A good illustration of the marked variation of exchange with orbital state has recently been given by Copland and Levy [30]. Using previously calculated wavefunctions for Co2+ they calculated the direct exchange for two Co2+ ions and found complex variations involving factors of more than f 2 in the different exchange constants. The superexchange case will not be qualitatively different, as shown recently by Hartmann-Boutron [31].

In practice, the variation of the exchange integrals with orbital state may lead to widely different amounts of anisotropy in the effective interaction. Thus in CoF, a dominance of the exchange involving the dy orbitals over the de orbitals appears to lead to a rather isotropic spin coupling [32-341, while in less symmetric cases such as a-CoSO, a large anisotropy seems to result from the competition [35]. For rare earths the situation is even more complicated and several attempts have been made at explicit superexchange calculations, using different physical approximations and fitting a number of parameters [36-401. So far only semi- quantitative agreement with experiment has been achieved.

IV. Form of pair interaction Hamiltonian within different single ion manifolds. - We now consider several combinations of the single ion states discussed in Sec. I1 to find expressions for the corresponding operators for the general pair interaction Hamilto- nians. Three of these have been considered extensively before [I, 31, and using the notation defined in Sec. I1 we can classify these as :

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these cases, the Hamiltonian can be expressed in terms of the ionic spins S, and S,, and it will have the general form of eq. (1). Qualitatively we would expect the cases with well isolated orbital singlets to be accurately represented by the leading isotropic term and this has been amply confirmed by experiment [I, 61.

Large anisotropies enter when we consider systems with unquenched orbital angular momenta. We can identify the following cases :

4) [Lt-L'] COUPLING. - This includes [L-L] and [L'-S] and [Lt-Q] as special cases. The appropriate operator can be expressed as [23, 41, 431

+ c GE~;

@(L;)

G$:(L;). [s, . s,

^-f-

+I ,

(3)

KK'QQ'

where X,,(MDD) is the usual magnetic dipole coupling, (p,

.

^p,)/r

^-

^3(p,

.

^r)

.

^(p,

.

r)/r with

Pi = h ( L i f 2 Si),

and the

G;(L)

are spherical tensor (Racah) operators which we may normalize as those defined by Smith and Thornley [44] and tabulated by Birgeneau [45].

The ranks of the tensors, k, k', K and K' are limited by several conditions, not unlike the corresponding crystal field operators : k, K

<

2 El and kt, K t 6 2 l,, where 1, and 1, are the angular momenta of the individual electrons (we consider here only one configuration of equivalent electrons), with k, k' and K

+

K' even.

Electronic multipole and virtual phonon exchange contribute to the second set of terms, while orbital exchange contributes to the third set. The parameters F::,' (of which there may be as many as 729 [23] for f-electrons) and G:$' (of which there may be 1225 [43]) are therefore only related in specific interaction models.

Since such models are quantitatively unreliable we must regard the F' s and the G' s as empirical parameters, not unlike the corresponding Slater integrals in the theory of atomic spectra.

Unfortunately there is no apriori reason for neglect- ing any of the terms, and in particular there is direct evidence that the higher degree exchange terms can be comparable to the lower degree terms, or even larger.

(See for example Ref. 45 and many of the references cited in Table I.) This is in marked contrast to the situation corresponding to eq. (I) in which the anisotropic and higher degree terms are orders of magnitude smaller than the leading term. The only significant reduction in the number of possible terms results from symmetry considerations (see below), or if one of the two ions has L = 0. In the latter case eq. (3) reduces to :

which is the expression first used by Levy [12] to ana- lyze the rare earth iron garnets.

5) [J-J] COUPLING. - Formally, this also includes the case Is'-s'] for general s'. The operator corresponding to this case can be obtained from eq. (3) using

also write it down phenomenologically

XI, =

4;;

G$(J,) @,:(J,) (5) where 1

<

2 1, 4- 1, and 1'

<

2 1,

+

1. For f-electrons there are 4096 parameters J!!: of which 1954 are independent. Nevertheless, we shall see that this equation is very useful in certain special cases, where the restriction 1, 1'

<

7 for f-electrons becomes very significant.

If we apply eq. (5) to a situation in wich J is replaced by an effective total angular momentum (ficti- tious spin) s', there is a further limitation 1

<

^{2 s;}^-t¹

and A'

<

2 s;

+

1. The special case of s' = 1 has been discussed in some detail by Elliott and Thorpe [14], and we treat the even simpler case of st = 1/2 next.

6) [(s' = 112)-(s' = 1/2)] COUPLING. - This case corresponds to the physically important situation of two weakly coupled Kramers ions, which includes many rare earth compounds. In principle one can again project the operator applicable to a larger mani- fold ( L or J) onto the states describing the effective spin s' [41, 471. However, in general this is not very useful since the most general operator coupling two sf = 112 systems

has only nine constants compared with the hundreds of the larger operators. In general all components of K may be comparable, including the antisymmetric terms [48].

The only cases in which it may be useful to relate the tensor K to the more complex operators are those in which one aims to compare interacions in different doublet states belonging to the same multiplet or where some special simplification is possible.

(See Sec. V below.)

7) [J-MI. - This is the situation in which a J- type ion is isolated in a strongly polarized environ- ment. Provided the direction of M remains fixed (along u say) we can simply replace all the operators on site 2 by their expectation values, and this simpli- fies the form of the Hamiltonian considerably :

This looks like a crystal field acting on ion 1 except that it contains additional odd terms, which can even split states with Kramers degeneracy.

8) [st-M,]. - For the case in which M is derived from an S-state we can readily extend this to allow for different directions of M,, and specializing to the particularly interesting case of interaction with an ion described by s' = 112 we find

A

El2 =

C

^{s;a Gap}^{M z p}^Y

a,#=x,y,s

(8) a form first proposed by Wolf [8] and verified experimentally by Wickersheim and White [I 1, 401.

V. Determination of interactions in real crystals.

- From the above it is clear that a complete analysis

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ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS C 1 - 29

TABLE I

Examples of Systems with Highly Anisotropic Interactions

System Type Mechanisms (") Expts. (b) Refs. (?

Co2 ⁺-Mn2 +(CoC12 .2 H20) s'-S (se) 9 70

C O ~ + - C O ~ + ( C O X , ( S O ~ ) ~ . 6 HZO) st-sf d, (se) 5, 6, 12 85

((Zn, Co) Cs3C15) st-sf d, (se) 1,3,5,6,13 52,53,86

(CoCL, . 2 H20) st-sf (se) 3, 5, 12, 13 62,87

(CoCI, .6 H20) sf-sf he) 3, 5, 6, 12 61, 64, 88

(a-CoSO,) sf-S' (se) 3, 5, 13 35

((Mi%, Co) F2) sf-S' (se) 1,2,3, 5, 8 32, 33, 34

Eu3 +-Cr3 ⁺(Eu3Ga5OI2) L-Q se 9, 11 67, 68, 77

(EuAlO,) L-Q se 11 76

Eu3 +-Fe3 ⁺(Eu3Ga5012) L-S se 9, 11 66, 68, 69

(Eu3Fe5012) L-M, se 10 72

Ho3+ -Fe3+(Ho3Fe501J J-M, se 2, 13 60,89

Er3+-Fe3+(Er3Fe,0,,) J-M, se 2, 3, 13 57, 59, 81

y b 3 + -Fe3+(Yb3Fe50,,) J-M, se 2, 3, 8, 13, 14 49, 81, 83

Ce3+-Ce3 ⁺((La, Ce) ES) (d) sf-sf d, (eq ?) (vpe ?) 1,5, 6 22,90

((La, Ce) C13) st-st d, eq, se 1 , 5 6 9 18, 25, 49

pr3+-pr3+(LaC13) st-sf d, (se) 1 55

Nd3 +-Nd3+((La, Nd) C1,) sl-S' d, (se) 1, 2,5, 9 46, 91, 92

(Nd ; CaF,] s'-sf d, be) 1 54

Eu3+-Eu3 (EuZO3) L-L se 5 71

Tb3 + -Tb3 + (Tb(OH),) s'-S' d, (se ?) 2, 5, 6, 7 20,93

((Tb, Y ) Sb) J- J (ie) 10 73

Dy3'-Dy3+(DyES) (*) st-st d 5, 6 63

(Dy3A15012) sf -st d, (se) 2 , 5 , 6 , 7 , 8 , 17,79,94

12, 14, 15

(DYPO~) sf-st d, (se) 2, 4, 5, 6, 7, 58

12

R~ ⁺-R3 ⁺(ROsz, RA12RIr2) ( e ) J-J (ie) 4, 5 19, 96

(RAl, RNi, R,Ni) ( e ) J-J (ie) 12, 13 97

u4 +-u4

⁺(Uo2) sl-st VPe 3. 8 29

(") Dominant mechanism (s), where identified : d = magnetic dipole, eq = electric quadrupole, vpe = virtual phonon exchange, se = superexchange, ie = indirect exchange via conduction electrons. ( ) indicates probable interaction mechanism.

(b) Numbers correspond to list given in Sec. VI. Only experiments which relate to determination of interactions are cited.

(3 Only some of the more recent references have been selected. References to earlier work may be found in these.

Cd) LaES = lanthanum ethyl sulphate, etc.

(e) R = rare earth.

will be possible in only the very simplet cases, and then only if a number of different experimental measme- ments can be made to determine the appropriate parameters. The role of the theory must be to reduce these parameters to a minimum, and this involves three different considerations.

1) LIMITATIONS IMPOSED BY SYMMETRY. - For the general Hamiltonians such as eq. (3) or ( 5 ) symmetry conditions contribute little, but for the simpler effective spin Hamiltonians such as eq. (4), (6) or (8) considerable simplifications are possible. Thus for the rare earth iron garnets Eq. (4) reduces to 10 terms [12] while the nearest neighbor coupling in the rare earth trichlorides and ethylsulphates involves only 2 parameters in eq. (6) [25]. Appropriate forms for the spin Hamiltonians in general cases have been discussed extensively [7, 8, 14, 23, 50, 511.

2) RANGE OF INTERACTIONS. - Apart from magnetic dipole coupling, it is very difficult to predict the varia-

tion of interactions with distance. It is generally assumed that most of the other interactions are fairly short ranged in character, so that an analysis fitting only the nearest and next nearest neighbors should be adequate. However there is now some empirical evidence [52] that non-dipolar interactions with more distant neighbors may not always be entirely negligible. Tn many cases, however, a short-range coupling should still be a good approximation.

One particular case in which long-range interactions are known to be important is in the indirect coupling via conduction electrons, but this is largely outside the range of our present discussion. (See Sec. VII below.)

3) IDENTIFICATION OF SIMPLE CASES. -One of the most important roles of theory is the identification of special cases in which the real interactions are in fact characterized by only a few constants. These constants can then be determined empirically to yield

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for further calculations of the detailed properties of the system.

One general class of simple cases are systems in which 32''' leads to equivalent effective spins st = 112 with relatively weak interactions. This immediately allows the use of eq. (61, and if the symmetry is fairly high a considerable reduction of the number of unde- termined parameters results.

A specially simple subclass of this case occurs when it can be shown that only one component of the K tensor is non zero for each pair of effective spins. One then has a true Ising-like system, albeit that the range of the interactions may be larger than that usually considered in Ising model calculations.

It is sometimes assumed that a sufficient condition for the reduction to the Ising case is an extreme anisotropy of the magnetic g-tensor. This will indeed reduce isotropic exchange and magnetic dipole coupling between the real spins to an Ising interaction between the effective spins [3], but it is not at all obvious that an operator of the general from of eq. (5) should have zero matrix elements between states for which those of J, happen to be zero. We shall discuss some parti- cular illustrations of this situation below (see Sec. VII).

VI. Experimental methods for determining interactions. - It is clear that the determination of a fairly large number of unrelated parameters requires a considerably more extensive range of expzrinnental methods than the relatively simple problem of finding one or two exchange parameters in the isotropic case [16]. In this section we shall list very briefly the methods which have already been used to study anisotropic interactions, but we must refer the reader to a number of original papers and the references cited in these for details of the sometimes very elegant analyses used to interpret the measurements.

1) EPR OF COUPLED PAIRS AND TRIADS [25, 33, 46, 53-55]. - This is a very direct method and it only suffers from the problem that isolated pairs or triads have a slightly different spatial relation than the corresponding pairs in the pure material. To some extent this difficulty can be overcome by comparing measurements in related systems.

2) OPTICAL SPECTRA IN ORDERED CRYSTALS [20, 49, 56- 601. - This also gives a direct measure of the interaction but it involves soms averaging over different neighbors.

3) COOPERATIVE MICROWAVE A N D FAR-INFRARED RESONANCES - These methods have only been used in a few cases involving highly anisotropic interactions 113, 35, 61, 621 but they can give very clear indi- cations of complex anisotropies.

4) MAGNETIC SUSCEPTIBILITY. - This is a common method but it is useful only if the susceptibility in the absence of the interactions,

x,,

can be estimated accurately. In that case one can estimate

A

in

as T -* oo and relate 1 to trace (Xij pP p;), where ,LL:

and tt; are the magnetic moment operators in the direction of the field [16, 18, 63, 641. In practice

x,

can often be complicated by large and unknown crystal

dent contributions.

5) VALUE OF T,. - The ordering temperature can only be related to the interactions in very simple systems and then only to about 10

%

accuracy [16].

But it is sometimes the only available method [19].

6) HIGH TEMPERATURE SPECIFIC HEAT. - For T 9 T, the magnetic specific heat varies as

CM = C2/T2

-+

C 3 / ~ 3

+

a > - ,

where C, can be related [18, 561 to trace (32;). In practice the separation C, from the measured total specific heat is difficult but this can sometimes be overcome by using a high frequency method [65].

7 ) MAGNETIC INTERNAL ENERGY. - The integral

CO

C, d T is equal to the total magnetic energy

0

and in a simple system this is proportional to the sum of all the interactions [17].

8) SPIN WAVE SPECTRA (inelastic neutron scattering, optical side bands and low temperature thermo- dynamic properties). - With complex interactions quite unusual spin wave spectra are possible [29], but with an explicit model a useful analysis is generally possible.

9) SECOND ORDER SHIFTS IN EPR SPECTRA. - These effects are quite complicated, but in suitable cases they provide very clear evidence of highly anisotropic interactions [66-701.

10) INDUCED MAGNETIC MOMENT AND INDUCED HYPERFINE STRUCTURE. - These are second order effects related to the above and they too have given clear evidence of anisotropic interactions [71-731.

1 I j SELECTION ^{RULES AND}ENERGY TRANSFER IN OP- TICAL SPECTRA.

-

The complex high rank interactions of eq. (3) and (5) clearly couple many electronic states not coupled by the more usual operator of eq. (1) [74, 771.

12) FIELD INDUCED SPIN-FLOP AND SPIN-FLIP TRANSI- TIONS.. - If the interactions are sufficiently weak, applied fields can significantly alter the ordered state.

The critical fields extrapolated to T = 0 OK can be related simply to the interactions [17, 62, 78, 791.

13) OBSERVATION OF THE UNUSUAL SPIN STRUCTURES

(elastic neutron scattering). - One significant conse- quence of anisotropic and antisymmetric interactions is the possibility of more complex (often non-collinear) spin structures than those possible with isotropic exchange [35, 80, 811. The interpretation is not always easy and it is necessary to note that the direction of the effective spin is not necessarily that of the ordered moment [821.

14) TORQUE MEASUREMENTS. - These give the angular variation of the free energy in the ordered state, but careful analysis is required [83].

15) SPIN-SPIN RELAXATION TIMES. - This is not a widely used method as the detailed analysis is too difficult. However, in highly anisotropic (Ising-like) cases, a very long relaxation time can provide evidence of the smallness if the non-Ising terms in XI, [84].

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ANISOTROPIC INTERACTIONS BETWEEN MAGNETIC IONS C l - 3 1

VII. Experimental evidence for highly anisotropic interactions. - Evidence for signijicantly anisotro- pic interactions has been accumulating for more than 10 years and there are now over 25 different examples in the literature. These are summarized in Table I. We have selected only cases where the evidence is clear-cut and there is a general concensus that anisotropic interactions are indeed important.

There are many other cases in which highly anisotropic coupling is surely also important but obscured by other complications, such as crystal field effects.

Several of the examples listed in Table I deserve special comment.

1. Yb3Fe50,,. - This was the first really clear- cut example of the failure of isotropic exchange. The complete difference in the topology and anisotropy of the magnetic (g) and exchange (G) 1491 tensors proves the existence of higher degree orbital exchange comparable with the isotropic part. Similar and more extensive results have recently been reported for Er3Fe,0,, [59].

2. CeCI,. - This is a very simple case which has been studied in considerable detail [25, 461. The ground state is a Kramers doublet well described by

I

J = 512, J, = IfI 5/2

> .

Correspondingly the magnetic g-tensor is highly anisotropic and one might expect similar anisotropy in the effective spin-spin interaction. For nearest neighbors this is actually what is found, but for next nearest neighbors there is an unexpectedly large and almost isotropic coupling. This can only be explained by therms of the form

in the general interaction Hamiltonian.

From a careful analysis of the EPR spectrum and a comparison with the similar case of Nd3+-Nd3+

coupling it has also been possible to demonstrate a significant contribution from electric quadrupole- quadrupole coupling, with negligible higher degree multipole effects.

3. ISING-LIKE SYSTEMS. - For the Ising condition to be valid we require ions with doublet ground states which are such that none of the general operators have matrix elements coupling the two states. One way this can happen is in rare earth ions with doublets described by

I

J, ^$.J:

>

^{with J:}

>

4. For such a doublet only operators @(J) with k 2 8 will have off- diagonal matrix elements, but the condition limits k to

<

7 for f-electron ions.

The best examples of this situation are Tb(OH), 1201 which has a ground state almost pure ( J , =

+

⁶

>,

and Dy,A1,Ol2 [94] and DyPO, [58] which have almost pure

I

J , =

+

¹⁵¹²

>

ground states.

A different type of Ising-like system is illustrated by CoCs3CI,. This has low lying states described by an orbital singlet (Q type) and S = 312. Within this manifold we would therefore expect almost isotropic exchange. However, the exchange is weak so that one must first consider a 12.4 OK splitting of the S = 312 state into two doublets [86], the lowest of which is

IS,= -f 312

>.

Within this doublet the isotropic S i . S j exchange now only has diagonal matrix ele- ments, and an Ising-like coupling between the efective spins again results.

It is interesting to compare this case with CeC1,.

In CoCs3C1, we have an isotropic coupling being projected into an anisotropic form, whereas in CeCl, an intrinsically anisotropic term happens to result in an almost isotropic coupling between the effective spins. These examples illustrate how careful one has to be!

4. SECOND ^ORDER^EFFECTS.- Several of the listed examples demonstrate the importance of second order effects, and it is important to note that the interaction effects are not limited to operating within any one manifold of single ion states. Indeed this is the origin of the well known d and A terms in eq. (I), but with low lying single ion states the effects can be very much larger.

In addition to second order effects involving the interactions twice, one can also have significant cross terms with other effects such as the Zeeman term.

These give rise to g-shifts [66] and induced moments [72, 731 and in any detailed analysis it is necessary to take these effects into account.

5. METALS AND INTERMETALLIC COMPOUNDS. -There is very little detailed evidence of anisotropic interactions in systems in which the interactions are mediated by conduction electrons, but complex effects still occur [97]. Evidence of higher degree interactions has been found in (Tb, Y)Sb [73], while Levy has recently shown [19, 961 that spin orbit effects in the conduction bands can give coupling terms like L,.S, in systems like ROs, and RIr,. Orbital effects have also been considered for the rare earth metals and their alloys [97, 1011, but there is as yet no detailled parametric theory. However, it seems quite clear that there will be many cases where interactions more complex than a simple S,.S, (or a corresponding 3,. J,) coupling is important, and further work on this problem will be of interest.

VIII. Conclusions. - It has now been establi- shed both theoretically and experimentally that the interactions betweenmagnetic ions will generally depend significantly on their orbital states and this can lead to large anisotropies in the effective spin-spin interactions. We must expect sizeable anisotropies whenever the single ions have more than one orbital state in an energy range comparable with the interactions, but no quantitative predictions are possible using only theory.

It is also possible to get higly anisotropic couplings when a weak interaction (even an intrinsically jsotro- pic one) operates only within a restricted manifold of states, which can project out certain (generally anisotropic) components.

Hamiltonian operators appropriate to the six most important cases are given. Detailed analysis of experimental data is possible only in cases where other complications can be removed and the problem is simplified by symmetry and a limited number of effective states per ion. In such cases it is possible to express the interactions in terms of a small number of empiri-

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cal parameters which can then be determined by com- Acknowledgements.

-

We would like to thank bining different experimental results. R. J. Birgeneau and P. M. Levy for a number of help- In all other cases involving orbital interactions only ful and stimulating discussions, and t o apologize qualitative analyses are possible and i t is meaningless t o all authors whose relevant work has been omitted t o fit data with a n arbitrarily simplified Hamiltonian. either through oversight o r lack of space.

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