INVESTIGATION OF MAGNETIC
INTERACTIONS OF
NICKEL (n), COBALT (H), AND MANGANESE (H),
IN MAGNETICALLY DILUTE SYSTEMS
INVESTIGATION OF MAGNETIC
INTERACTIONS OF
NICKEL (n), COBALT (H), AND MANGANESE (H),
IN MAGNETICALLY DILUTE SYSTEMS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS Ir. H. B. BOEREMA, HOOG-LERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET
COLLEGE VAN DEKANEN, TE VERDEDIGEN OP WOENSDAG 29 MEI 1974 TE 16.00 UUR
DOOR
GERARDUS
CORNELIS MARIA VAN LEEUWEN SCHEIKUNDIG INGENIEUR j GEBOREN TE LEIDSCHENDAM ' '
1974
DRUKKERIJ J. H. PASMANS. 'S-GRAVENHAGE
BIBLIOTHEEK TU Delft P 1851 4044
C 577830
Aan mijn ouders Aan Joke
CONTENTS
CHAPTER I INTRODUCTION 1.1. Definition of problem
1.2. Direct and superexchange interactions of transition metal ions
1.3. Outline of this thesis
7 7
8 13
CHAPTER II MAGNETIC INTERACTIONS IN BULK COMPOUNDS 15
2.1. Introduction 15 2.2. Nearest neighbour interactions only 16
2.3. Nearest and next nearest neighbour interactions 18 CHAPTER III CRYSTAL STRUCTURES AND STATISTICS OF CLUSTERS 20
3.1. Cluster statistics in the olivine structure 20 3.2. Cluster statistics in the MgO structure 22 3.3. Cluster statistics in perovskite structures 24 CHAPTER IV MAGNETIC PROPERTIES OF COUPLED SPIN SYSTEMS 26
4.1. Introduction 26 4.2. General Hamiltonian 26 4.3. Two-spin systems 27 4.4. Three-spin systems 33 4.5. The biquadratic exchange term 34
CHAPTER V BIVALENT NICKEL IN MgO AND UMgP04 36 5.1. Theory of the ground state of octahedral Ni** 36
5.2. Formulae for cluster susceptibiUties 39
5.3. Experimental 41 5.4. Results and discussion 42
CHAPTER VI BIVALENT MANGANESE IN MgO AND UMgPO* 6.1. Introduction
6.2. Formulae for cluster susceptibilities 6.3. Experimental
6.4. Results and discussion
SO SO
so
S2 S2CHAPTER VII BIVALENT COBALT IN MgO AND LiMgPO* 7.1. Theory of octahedrally co-ordinated Co^* 7.2. Single ion and cluster susceptibilities 7.3. Correction for doublet-quartet interactions 7.4. Experimental
7.5. Results and discussion
57 S7 S8 61 62 62
6
CHAPTER VUI MAGNETIC PROPERTIES OF Co^*, Mn^"^ AND Ni^* IN
MgS 71 8.1. Introduction 71
8.2. Preparation of the samples 71 8.3. Results and discussion 73 CHAPTER IX CONCLUSIONS AND COMPARISON TO BULK DATA 80
9.1. Exchange parameters, derived from clusters 80 9.2. Exchange parameters, derived from bulk compounds 80
9.3. Comparisons and conclusions 81
9.4. The sulphides 84
SAMENVATTING 86
CHAPTER I
INTRODUCTION
1.1 Definition of problem
Two main problems in the theory of magnetism of transition metal compounds can be stated as follows:
a) What is the nature of the coupling between two spins, localized on different atoms? b) Given an interaction of certain type, how are the macroscopic properties
calcu-lated?
Problems of type b) have been discussed thoroughly in several standard works on magnetism [1,2]. In these calculations nearly always interactions of type
2 J S r S 2 (1.1) between pairs of spins Sj and S2 are assumed. Though the Curie-Weiss temperature 6
can be expressed in terms of the parameters J and the spin quantum number S through exact formulae, the relationship between the Neel temperature (J^.) and these para-meters is strongly dependent upon the model used. Unfortunately, the magnitude of the interactions are rarely known explicitly, so that the validity of a model cannot be checked independently. This is the first reason why direct determinations of the interactions between spins, through measurements on magnetic ion pairs in diamag-netic host lattices, are actually needed [2].
From a physical and theoretical point of view the question about the nature of the coupling is perhaps even more interesting. Since Anderson's theory of superex-change, this phenomenon seemed to be rather well understood qualitatively [3]. However, in some cases the agreement between theory and experimental data was far from satisfactory, while results of quantitative calculations based on Anderson's theory seldom differ less than a factor two [4, 5,6] from the experimental values. The reason of these discrepancies is frequently, that exchange parameters are ob-tained as small differences of large numerical quantities. Thus, the phenomenon of superexchange is very compUcated and reliable experimental data from which a good approximate theory might be built up, will be wellcome. Against this background we shall be concerned to build up a framework of experimental data by studying the magnetic superexchange interactions of three related transition metal ions, Ni, Co and Mn, in the simple fee lattices of MgO and MgS. The results will be compared with the predictions of a qualitative scheme, which is based on the combined theo-ries given by Anderson-Owen and Van Vleck-Tanabe [3, 7 , 8 , 4 ] .
8
Experimental techniques, being of interest for a study of the magnetic ion pairs are:
— e.p.r. spectroscopy
— optical absorption and fluorescence — magnetic susceptibility measurements.
With the e.p.r. experiment only very small interactions can directly be deter-mined from a spectrum. Higher interactions p * 0.5 K) can roughly be deterdeter-mined from the temperature dependency of the signal. A good survey of the method is given in reference [9].
High coupling constants may be obtained with reasonable accuracy from opti-cal data of ions in the solid state giving sufficiently narrow bands; the R bands (^A2—^E transitions) of octahedrally co-ordinated Cr^ are good examples [10—14].
A good method of obtaining interaction parameters in a broad range (1 K — 100 K) is measuring the magnetic susceptibility of clusters. Effectively energy levels of cluster states are measured, which can directly be connected to the exchange coupling constant. The results may be very reliable if there is only one type of inter-action present or - in the case that there are more interinter-actions - if the interinter-action constants differ by a factor 3 or more. This method will be worked out thoroughly in this thesis. Previous investigations to relate the magnetic properties of diluted compounds to the exchange coupling constants have been performed on Co^ in KMgFs [15] andMn^inCdS [16].
1.2 Direct and superexchange interactions of transition metal ions
The choice of the magnetic ions is determined by the realization that the mag-nitude of the superexchange interaction is strongly governed by the types of d orbit-als occupied by unpaired electrons. In Ni^, Mn^ and Co^, with the configurations (t2g)*(eg)^, (t2g)^(eg)^ and (t2g)^(eg)^, a half filled shell of Cg-electrons, a half filled shell of Cg and t2g electrons and a half filled shell of eg and a partly filled shell of t2g electrons are involved respectively.
In order to understand the implications for the magnetic behaviour, a short in-troduction to the approach outlined by Owen will be given here.
Let us consider two electrons in ds^^ ^2 orbitals, both having overiap with a Pz-orbital of an oxygen ion, lying in the middle of the axis connecting the two d-ions (Fig.(I.l)).
dZD CXD etD
oAn orthogonal localized set of orbitals can be obtained by mixing each of the d-orbitals a little with the oxygen 2pz orbital and the d-orbital of the other d-ion and by mixing the p^ orbitals with the d-orbitals.
V'i = - d 3 z ' - r ' + apz+^d5,2_,2 (1.2) <p2=d5z2-r2+ap,-)3d|,2_^2 (1.3) V'p = Pz-7d3z2_r2+7d5z2-r2 (1.4) The ground state of this four electron system can be written as:
* 0 = IVl<^p'iPp<^2 I (1-5) So far * o is still fourfold degenerate and composed of a singlet and a triplet. If i^i,
1^2 and (/)p are orthogonalized, the Coulomb repulsions lead to a lowering of the fer-romagnetic (triplet) state relative to the antiferfer-romagnetic (singlet) state, which is governed by the so-called 'Heisenberg exchange'. Hence in this approximation only ferromagnetic interactions would occur.
Configuration interaction changes this picture, particularly by the effect of those configurations, that only occur as singlets. For the chosen basis set two such configurations exist:
* i = I'/'i VJii^pi^p I (1-6) * 2 = liPj V'2 iJip i^ I (1-7) These are the so-called ionic states, since both d-electrons are localized on one of the
d-ions. Configuration interaction now results in lowering the singlet *o(S) (Fig. 1.2). If the configuration interaction is sufficiently strong the antiferromagnetic contribution dominates the ferromagnetic one. Quantitative quantummechanical calculations based on this model are rather laborious and hence rather rare. Quali-tatively this model of superexchange leads to the prediction that the antiferro-magnetic interaction between an electron in orbital i^a. localized on atom a, and an electron in orbital ip^,, localized on atom b is growing stronger as both orbitals have more overlap with a same intervening ligand orbital.
It will now be clear, that the exchange interaction will be dependent on the configuration of the cations with respect to the intervening anion. Thus for instance 90° and 180° exchange paths can be distinguished. A qualitative scheme of inter-action strengths for 90° and 180° superexchange paths of transition metal ions are given in Table I-l and Fig. 1.3, where the direct (ferromagnetic) contributions are neglected.
10 ',(s) yv,(st YJ(! YoS) Yo<T) Y " ® ' / / \ Yo<T) ^ Yo<T) \ \ \ \ \ \ \ \
\ T»
(S) Hcisenberg connguratton Direct ochanga InlaractionFig. 1.2. Mechanism of superexchange in insulators.
individual d-electrons. The exchange energy, coupling two ions is given by the sum of coupling energies between all pairs of electrons (the two electrons in a pair belong-ing to two different cations), i.e., we have
V,2 = 2 2* 2 J..S.-S. (1.6)
where s. = spin of electron i on atom a s. = spin of electron j on atom b
N = number of unpaired electrons on atom a N. = number of unpaired electrons on atom b
I.. = exchange coupling constant between an unpaired electron i (in orbital ^p.) and an electron j (in orbital ip.).
(Possible anisotropics in the interaction nave been neglected here.)
iz;
180° interactions 9 0 ° interactionse g - e g
C$3
GXE)
6 8 3
C$3 GX3
CS3
d s j J - r J — d 3 l ' - r - dsz' - r' — d j i J . r i tjfl - t j j <l»i-tlx d v i - d eg _ tag5 0 G
etc GXD
d 3 2 > . , ) _ d , d j j ^ - r . - d ,Fig. 1.3. Orbital interactions for 180 and 90 paths.
In the theory of magnetism (1.6) is always rewritten in the form of (1.7)
V.2 = 2 J 3 b S , - S , (1.7) where S^ and Sj^ are the total spin quantum numbers on atom a and b respectively.
12 Table I-l
Estimate of interaction strengths between different types of orbitals. Only the inter-actions, which are expected to be non-zero have been listed.
type of interacting orbitals
a)eg-eg: ds^z.^^-dsz^-r^ b)t2g-t2g: d x z - d x z dyz-dyz dyz -dxy C)t2g-eg: d3z2_^2_dxz d x z - d x ' - y ' 90° superexchange overlap a.0 7T-0 nO TT-ff estimated interaction zero zero zero weak medium medium 180° superexchange overlap a-a 7T-0 TT-O TT.O estimated interaction strong weak weak zero zero zero For the case that the atoms have no orbital degeneracy in their ground state Van Vleck has shown, that it is justified to describe the magnetic interaction by Eq. (1.7). The relation between J ab and i.. is than (8):
Jab=: 1
N
a
2 2 J
NaNb i=I j=l
This relation has also been found by Tanabe [4]. Transition metal ions, where the interaction can be described by Eq. (1.7), are for example octahedrally co-ordinated V2^Cr^Mn2*andNi^.
For ions with an orbitally degenerate ground state Sa and Sb are no longer good quantum numbers. It is now desirable to describe the interaction on the basis of the fictitious spins Jg and Jb of the ground multiplet.
V,; 2 Ku J ab 'ab (1.9)
The relation between Jab and Jy is now given by: Jab - 2; Cjj Jjj
•J
(1.10) The coefficients Cy can be calculated from the wave functions of the ground state.
From Table I-I and the Equations (1.8) and (1.10) it may be concluded, that, going from octahedrally co-ordinated Ni^ (having unpaired electrons in the two Cg orbitals only) to Mn^* (having unpaired electrons in the two eg orbitals as well as in the three t2g orbitals) the 90 superexchange interaction should considerably in-crease relative to the 180° interaction. The Co^* ion should show a behaviour inter-mediate between the other two ions.
It is interesting now to compare these predictions with the data derived from the magnetic properties of NiO, CoO and MnO by molecular field analysis (see Table 1-2):
Table 1-2
Predicted and observed J (90°) and J (180°) ratios for NiO, CoO and MnO [1,2]. The predicted values have been derived according to the 'order of magnitude calcu-lations' given by Owen [7]. J '= J (90°)/J (180°).
compound author NiO Perakis Smart CoO Singer Kanamori MnO Foex J(90°) K 150 50 1 6.9 7.0 J (180°) K 90 85 20 21.6 3.5 J ' observed 1.7 0.6 0.05 0 3 2 J ' predicted 0.01 0.1 1 It is seen, that predictions and experimental data are inconsistent. It is the main purpose of this thesis to provide independent experimental data for J (180 ) and J (90°) for Ni^, C o ^ and Mn^. Then it may be decided whether these data correspond to the predictions of Table 1-2. If so, estimates of interaction parameters by the molecular field method turn out to be seriously in error. If not, the theory of superexchange interaction should be revised.
1.3 Outline of this thesis
The investigations of this thesis will be performed by incorporating the transi-tion metal ion under consideratransi-tion in diamagnetic host lattices with structures, iso-morphous to those of the bulk magnetic compounds, and determining the interaction parameters from measurements on magnetic ion pairs, statistically occurring in the diamagnetic lattice. First the systems NixMgi _xO, Co^Mgi _xO and MUxMgj _xO will be considered, where 90° and 180° interactions can be studied. Supplementary experimental data about the angular dependency of superexchange interactions may be obtained from a study of systems having ohvine structure (for example
LiMxMgi -XPO4), where 135 superexchange paths occur as main interactions. A succinct survey of the classical methods for deriving interaction parameters from the magnetic properties of bulk compounds is given in Chapter II. Statistics for the occurrence of the various types of ion clusters in diamagnetic host lattices with randomly distributed paramagnetic ions are given in Chapter 111. In Chapter IV the magnetic properties of such clusters are discussed. The 90°, 135° and 180° superexchange interactions of Ni^ ions over 0^" ligands are presented in Chapter V. In Chapters VI and VII similar investigations are described for the ions Mn^ and Co^ respectively. Results on homogeneous sulphides of the type NixMgi _xS,
14
CoxMgi_xS and MnxMgi_xS are presented in Chapter VIII. In Chapter IX a general discussion of all results will be given against the background of data on bulk compounds available in hterature.
References
l.G.T. Rado andH. SM(Ed.), Magnetism 3, 2, 63 (1965).
2. F. Seitz and D. Tumbull (Ed.), Solid State Physics 14,99 (1963). 3. P. W. Anderson, Phys. Rev. 115,2 (1959).
4. K. Gondaira and Y. Tanabe, J. Phys. Soc. Japan 21,1527 (1966). 5.A^.Z,. Huang and^. Orbach, Phys. Rev. 154,487 (1967).
6.D.E. Rimmer, J. Phys. C 2, 392 (1969).
l.J. Owen, Proc. Neth.-Norw. Reactor School, p. 127 (1964).
8./.^. van Vleck, Rev. Univ. Nacl. Tucuman ser. A (Argentina), 14, 189 (1962). 9. J.C.M. Henning, Ned. Tijdschr. Natuurk. 35, 317 (1969).
10. K. W. Blazey and G. Burns, Phys. Letters 15,117 (1965). 11. A.E. Nikiforov, Sov. Phys.-Solid State 8,1340 (1966). 12. A". H^. Blazey, Solid State Comm. 4,541 (1966).
U.K.W. Blazey and G. Bums, Proc. Phys. Soc. 91,640 (1966).
14. L.F. Mollenauer and A.L. Schawlow, Phys. Rev. 168,309 (1968).
IS.MM. Kreitmann and J.G. Daunt, ibid., 144, 367 (1966).
CHAPTER II
MAGNETIC INTERACTIONS IN BULK COMPOUNDS
2.1 Introduction
In magnetic bulk compounds the Ne'ël- (Curie) point,, T^ (Tc), the Curie-Weiss temperature, Ö, and the magnetic susceptibility at the Néël point, X(TN), are the most important sources of information about magnetic interactions.
From the temperature dependency of the magnetic susceptibility, the Curie-Weiss temperature is determined by extrapolating the high temperature assymptote of the Xm vs. T curve to Xm - 0. This is illustrated in Fig. II.I, where the reciprocal susceptibihty as a function of temperature is shown for a ferromagnetic and an anti-ferromagnetic compound.
The Néël point can be determined from magnetic susceptibility measurements (see Fig. II.I) or, more accurately, from specific heat measurements.
Fig. II.1. Reciprocal susceptibility as a function of temperature for a) fenomagnet, b) antiferro-magnet.
16
The calculations of Ö, TN and X(TN), available in literature, are based on the Heisen-berg-Dirac model, which has as its basis the assumption that the interactions between a pair of magnetic atoms i and j is:
K = 2 Jij Si Sj (2.1)
where Jij is the exchange interaction between the i-th and the j-th atom and Sj and Sj are spin operators. Thus the Hamiltonian for a system of N atoms is:
JAj = 2 2 2 Jij Si • Sj (2.2)
i ^ J = l
If the system is a crystal, which contains only one type of magnetic atom, it is pos-sible to take advantage of the inherent symmetry and rewrite the Hamiltonian in the form:
Mij = 2 2 Jk 2 Sj • Sj (2.3)
k=l k-th neighbours
The sum over k is over sets of neighbours, from the nearest to the most distant, while the second sum is over all pairs within the set of k-th nearest neighbours. As the magnitude of interaction rapidly decreases with distance, in practice there are many cases where only one type of neighbour is taken into account, while very few cases are known, where more than two neighbour types are involved.
Using this Heisenberg model, the relation between 6 and the interaction para-meters Jj can be derived as:
0 = _ •2S(S+I) 2 7 J (2.4) 3k i
where Zj is the number of neighbours of one ion with interaction J;. Though relation (2.4) is exact, the calculation of TN and XmCTw) requires the solution of a coopera-tive problem and hence some form of an approximation. The discussion of these cal-culations rather naturally divides into two parts. First we shall consider the case where the nearest-neighbour only limitation can be applied reasonably well and secondly the case in which magnetic compounds have two or more sets of interactions. In the first case refined theoretical approximations are available, while in the case of more inter-actions only the qualitative molecular field analysis can be applied. Therefore the results obtained in the latter case are basically less reliable than those obtained in the first one.
2.2 Nearest neighbour interactions only
If interactions with only one set of neighbours (generally the nearest) are im-portant, the interaction parameters can be derived with good reliability from the magnetic properties of bulk compounds. The molecular field method, as simplest
ap-proximation, can be applied at all temperatures, but is wrong in some details. The Hamiltonian for general spins Sj is [ 1 ] :
5f = 2 2 2 Jij Si • Sj + 2 g|3H • Si (2.5)
i ^ j i=l i
The basic idea of this method is now to replace all spins except the one under consid-eration by their averages and treat the statistics of this one spin alone. This requires that we solve the problem of the effective Hamiltonian for spin i:
3(i=Si[4 2Jij<Sj> + g/3H] (2.6) j
The Umit in which this becomes exactly valid is that of long-range forces in which the number of neighbours Z goes to infinity, since then the fluctuations of the sum will become negligible. The results of the M.F. theory in this case are:
T N = ^ ^ ^ t l l . Z . J (2.7) and
x " J i ( T N ) = ^ | f ^ Z . J (2.8) where Cm is the Curie constant and k is the Boltzmann factor.
Improved results can be obtained by using other approximations, which are more precise but are valid only in a limited temperature range. Thus a high temper-ature approximation such as the Bethe-Peierls-Weiss (B.P.W.) method takes account of short range order effects, which are neglected in the M.F. treatment. The B.P.W. theory can be compared most easily with experiment by using the general relations:
I N = 0(Z,S)J/k (2.9) and
x"Ji(TN) = Z(Z,S)J/kCm (2.10) In these equations 0(Z,S) and Z(Z,S) are calculated numerically from a given
parti-tion funcUon for a B.P.W. cluster of a center atom and its Z nearest neighbours. A table of B.P.W. values of 0 and Z for Z = 6 and 8 and S = ' / 2 , 1, '/z, 2, ^/2, is given in reference [2].
Together with the formula for the 0 temperature,
- e = 2 S(S+1) . Z . J (2.11) 3 k ^ ^
18
some idea of the internal consistency of the Heisenberg model. Unfortunately, the B.P.W. method cannot be applied to the fee lattice.
It should be noted, that the most general interaction between two spins is written as
JC = S i / S j (2.12) where / is a tensor. If this tensor is isotropic the interaction can be written in the
simple Heisenberg form. In the Ising model it is assumed that the interaction tensor only contains the J^z element. The Ising type of interaction is therefore given by:
3f = JzzSzSz (2.13) Though the Ising model is useful in some cases, for the materials studied in this
thesis it is not relevant.
2.3 Nearest and next-nearest neighbour interactions
In the majority of the magnetic compounds, the next-nearest neighbour action is equally important or even more important than the nearest neighbour inter-action. Though some attempts to extend the more advanced B.P.W. and Danielian-Stevens treatments to systems having two or more interactions have been made, the only practical way of handling such systems is the molecular field treatment. We shall consider cases where there are only two important interactions with the parameters Ji and J2. As a result the M.F. analysis expresses the three numerical observations TN,Ö and Xm(TN)as linear combinations of the parameters J, and J2. Though it
might look, that any two of the measurements would be sufficient to determine Ji and J2, it should be noted that TN , Ö and Xm(TN) are interdependent according to the M.F. theory:
T N - Ö = - % - (2.14) Xm(TN)
For fee lattices and for different kinds of orderings the relations between J1, J2 and ö,TNandCm/Xm(TN)are[3]: type 1 of ordering: —41, + 6J2 = T N 2 - I 6 J 1 =Cm/Xm(TN)2 12J,+6J2 =0 2 type 2 of ordering: - 6J2 = TN 2 - 1 2 J , + 1 2 J 2 =C„/Xm(TN)2 I2J,+6J2 =6 2
type 3 of ordering: - 4Ji + 2J2 = TN 2
- 1 6 J , + 4 J 2 =Cm/Xm(TN)S I 2 J 1 + 6 J 2 =0 2
where
2 = - 3k/2S(S+l)
In the compounds NiO, CoO and MnO type 2 of ordering has been found. From each of the three possible pairs of equations a set of J j , J2 values is ob-tained. In order to decide which set of J j , J2 is the most reliable one, the following considerations should be made:
a) The formula for 0 is exact, while those for TN and Xm(TN)-Cm are M.F. approxi-mations.
b) The experimental error in 0 is often considerable, especially when high tempera-ture susceptibihty measurements are needed in order to determine a reliable assymptote.
Though in many cases the three sets of (Ji J2) values agree within 30%, for the case of NiO the internal consistency is much poorer so that even the right order of mag-nitude of the J i , J2 parameters thus obtained may be doubtful. In this connection it should be noted, that the molecular field theory is expected to become less reliable as S decreases.
References
1. F. Seitz and D. Tumbull (Ed.), Sohd State Physics 14, 116 (1963). 2. J.S. Smart, Phys. and Chem. Sohds 11,97 (1959).
20
CHAPTER III
CRYSTAL STRUCTURES AND STATISTICS OF CLUSTERS
In magnetically dilute crystals the susceptibihty is calculated by considering the contributions from single magnetic ions as well as the contributions from various aggregates of these ions. Statistics for the occurrence of the various types of clusters formed by paramagnetic ions, randomly distributed in diamagnetic host lattices, will be given for the ohvine structure, the MgO structure and the perovskite structure. 3.1 Cluster statistics in the olivine structure (LiMgj _xMxP04)
LiMgP04, LiNiP04, LiCoP04 and LiMnP04 all have the orthorhombic olivine structure. In this structure the oxygen ions form a slightly distorted hexagonal close packing. The Mg^* and the Li*^ ions are surrounded by distorted octahedra of oxygen ions, but occupy different sites in the lattice. Each of the Uthium ions lies at a center of inversion ('i' site), while the magnesium ions occupy sites of mirror symmetry ('m' sites) and Ue in puckered planes, perpendicular to the a-axis. (See Fig. III.l)
In the following, paramagnetic ions (Ni^, Co^*, M n ^ will be designated with the symbol M^*.
Neglecting in a first approximation the superexchange interactions between M^* ions in different puckered planes (M—O—P—O—M paths), it can easily be seen, that each M^* ion may interact with four neighbours in the same puckered plane through a 132° M—O—M superexchange path. Assuming a random distribution of the M^ ions over the bivalent cation sites, the numbers of bi- and trinuclear clusters have been calculated as follows. Let the number of bivalent cation sites be N and the fractional concentration of M^"^ be x. Then, since the probabihty that a bivalent cation site be occupied by an M^* ion is x, the probability that a M^* ion has no M^* neighbours is equal to:
P i = ( l - x ) ' * (3.1) The total number of single ions (S) is than:
N s = N x ( I - x ) ' ' (3.2) Similarly the number of binuclear clusters can be derived by calculating the
proba-bility (P2) that an M^* ion is surrounded by just one other M^* ion:
000 0.00 000 © 0 2 5 00.25 @ 2b Fig. Ill.la O no.75 00.75 00.75 O ®0.00 ®0.50 ® * (DO 25 00.25 O0 25 O 0075 ©0.75 O O '•0.50 «0.50 «O.SO • ©0.25 00.25 © O 00.75 O 0.75 O O ®0.50 ®0.00 ® ® QO.25 00.25 00.25 O 00.75 ©0.75 O ® 000 Oj)0 0^0 ^ Fig. m.lb
ó
ó
o
• Li o P "• ® Mg/M O OFig. III.la. Projection of idealized LiMgP04 unit cell along the o-axis. b. 'Puckered' plane perpendicular to a-axis.
22
The number of binuclear clusters is now: NBIN = - Nx.p2 = 2 Nx^ (1-x)^
2
Formulae for the number of single ions Ng, binuclear clusters Ngi^ and various tri-nuclear clusters are given in Table III-I, together with some numerical data.
Table III-1 a
Formulae for the number of different types of bi- and trinuclear clusters per mole ofUMxMgi_xP04(a = 4).
type name number Ni(x) single S Nx-(l-x)'* binuclear BIN Nx • H ax • (I -x)* trinuclear TRI Nx •'/4ax • 3 x - ( I - x ) '
Table Dl-lb
Numerical data on Pi(x) = n Ni(x)/Nx = fraction of M^ ions involved in a cluster of type i. (n = 1,2, 3 for single ions, bi- and trinuclear clusters respectively.)
X 0.01 0.02 0.03 0.04 0.05 S 0.9606 0.9224 0.8853 0.8494 0.8145 BIN 0.0376 0.0709 0.0999 0.1252 0.1470 TRI 0.0018 0.0061 0.0127 0.0208 0.0299 TOTAL 0.9999 0.9994 0.9980 0.9954 0.9914
3.2 Cluster statistics in the MgO structure
The structures of MgO, NiO, CoO and MnO are all of rock salt type. From Fig. III.2 it can be seen that each cation may be connected to twelve nearest neighbours through 90 ('A') superexchange paths and to six next nearest neighbours through a 180 ('B') superexchange path. A list of all possible types of bi- and trinuclear clus-ters is given in Fig. Ill .2. Though cluster statistics are much more comphcated this time, the calculational procedure is in principle the same as in Section 3.1. Assuming again a random distribution of M^ ions over the Mg^ sites, the number of each pos-sible type of binuclear and trinuclear M^-cluster in MxMgj _xO per mole M^* has been calculated. The results are given in Table III-2a and III-2b. The supplement to 1.0000 in 'total' is the fraction of M^-ions included in clusters of four or more.
^ ^ ^ ^ Magnesiumoxyde
n
A path 90° superexchange B path 180 superexchange ( 3 oxygen A moonesium Fig. III.2a type type^u
r%
- o 5a) Ó < • Q BINA BINB TAAA Ö • W A A 5b) - go c ^ ^ J
o—i—o
7a) 7b) 8a) 8 b) ^ " ^ TABA TAB > TBB Fig. III.2bFig. III.2a. Structure of magnesiumoxyde.
IA
Table ni-2a
Formulae for the numbers of different types of bi- and trinuclear clusters per mole of MxMg|_xO; a= 12, b = 6, x = fraction of cation sites occupied by M ^ .
type 1 2 3 4 5a, b 6 7a, b 8a, b name S BINA BINB TAAA TAA TABA TAB TBB number N(x) N x ( I - x ) ' » N . ' / 4 a x 2 ( I - x ) " N.V4bx2(I-x)3° N ^ a x 2 . * / 3 x ( I x ) -N.'/4ax2.5x(I-x)3'* N-'/dax^ •2x(l-x)3'* N - ^ a x 2 - 8 x ( l - x ) 3 8 N-'/4bx^ •5x(I-x)'*' Table lll-2b
Numerical data on Pi(x) = n Ni(x)/Nx = fraction of M^ ions involved in a cluster of type i (n [ 1,2,3 for single ions, bi- and trinuclear clusters respectively).
X S BINA BINB TAAA TAA TABA TAB TBB TOTAL 0.0050 0.9137 0.0527 0.0258 0.0005 0.0019 0.0008 0.0030 0.0009 0.9993 0.0100 0.8345 0.0924 0.0444 0.0018 0.0064 0.0026 0.0098 0.0030 0.9984 0.0150 0.7618 0.1215 0.0572 0.0034 0.0121 0.0048 0.0182 0.0054 0.9846 0.0200 0.6951 0.1419 0.0655 0.0051 0.0181 0.0072 0.0267 0.0079 0.9676
3.3 Cluster statistics in perovskite structures
In some special cases the 'A' type interactions in the MgO structure may be neglected. It is then obvious that each magnetic ion may be connected to only six neighbours through 180° paths. In this case the type of clusters and their concen-trations can be taken identical to those to be expected for the simple perovskite structure. Formulae for the various numbers of binuclear (BINB) and trinuclear (TBB-L for linear clusters, TBB-H for angular ones) types are given in Table III-3.
Table ni-3a
Formulae for the numbers of various types of bi- and tri-nuclear clusters per mole of MxMgi _xO, taking into account 180° interactions only
type 1 3 8a 8b name S BINB TBB-L TBB-H N(x) * Nx(l-x)* 3Nx2 ( l - x ) * ° 3Nx3 (I-x)»'* 12Nx3(I-x)'3 Table III-3b
Numerical data on the fractions P(\\
X S BINB TBB-L TBB-H TOTAL 0.0100 0.94148 0.05426 0.00078 0.00316 0.99968 0.0200 0.88584 0.09805 0.00271 0.01107 0.99768 0.0300 0.0400 0.83297 0.78276 0.13274 0.15956 0.00529 0.00813 0.02181 0.03388 0.99280 0.98433
26
CHAPTER IV
MAGNETIC PROPERTIES OF COUPLED SPIN SYSTEMS
4.1 Introduction
It has been pointed out in Chapter I, that we shall in the first place be concern-ed with obtaining information about energy levels in pairs of magnetic ions. These energy levels may be related through an exact formula to the components of a mag-netic interaction tensor in the spin Hamiltonian, describing the lower cluster states. In this chapter we shall investigate what types of spin Hamiltonian terms will be required to describe clusters of different symmetries. Furthermore we shall show, how the magnetic properties of coupled spin systems can be derived from the eigen-values of these spin Hamiltonians.
4.2 General spin Hamiltonian
Our problem can be stated as follows: given a polynuclear system, containing n magnetic ions, each of whose ground states can be described by an effective spin Hamiltonian K,, and given a certain type of interaction Pjj between each pair of ions ij, how do we calculate the magnetic properties of the total coupled spin system?
If the thermally occupied states of each separate ion i can completely be des-cribed by Ki, i.e., if the ion i has one thermally occupied multiplet, then the thermal-ly occupied states (which are the states we are interested in) of the coupled spin sys-tem can be described by the following spin Hamiltonian [1]:
j(; = 2 3(; + E Pij (4.1)
i i < j
This is the case for ions such as octahedrally coordinated V^*, Ni^"*^ and Cr^, tetra-hedrally coordinated Co^* and both tetratetra-hedrally and octratetra-hedrally coordinated F e ^ and Mn^. However, ions, which have orbital degeneracy in their lower crystal field term (in for example octahedrally coordinated C o ^ usually have several thermally occupied multiplets. In this case the coupled spin system shows excited levels arising from interactions between ions, where one or both are in an excited multiplet state. Each category of these excited pair levels should be described by an-other spin Hamiltonian.
4.3 Two-spin systems
4.3.1 Spin Hamiltonian and basic functions
For a single magnetic ion the spin Hamiltonian can be written as:
3 ( ; = S D S + H G S (+ higher order terms) (4.2) If we include magnetic interaction, the spin Hamiltonian describing the lower
elec-tronic levels of an isolated pair of magnetic ions is [2]:
3Q = SiI>, Si + H G i Si +S2£»2 S2+HG2S2 + S i / S j (4.3) (+ higher order terms)
where we have used the following symbols: Si, S2 real or effective spins of ion 1 and 2
Di,D2 : zero field splitting tensors G\,G2 : g-tensors
ƒ : magnetic interaction tensor |3 : the Bohr magneton H : the appUed magnetic field.
The basisfunctions we shall use are the total spin functions IS, M>, which are formed by vector coupling of Si and S2:
IS,M>= S C(SiMiS2M2SM) ISiJVl, > IS2,M2> (4.4) M1M2
where IS1, M1 ) and IS2, M2) are the spin functions of the ions 1 and 2 and c(SiMiS2M2SM) are the Clebsch-Gordon co-efficients. The IS, M ) functions trans-form according to the irreducible representations D* from the direct product:
S 1 + S 2
D^ ® D'2 = 2 D' (Si > S2) (4.5)
S=S 1 — S2
in the full rotation group 0(3).
In the absence of a magnetic field the matrix of JC is diagonal if the spin Hamiltonian does not contain any zero field splitting tensors and if the J tensor is isotropic. In this case the system behaves as if its symmetry was 0(3) and the Is, M) states belonging to a same cluster multiplet are still degenerate. In reality, however, the highest symmetry of a system formed by the ion pair and its neighbouring atoms is an axial one [2]. If the system has the symmetry G C 0(3), the multiplets Dj spHt according to their subgroup structures,
28
where r | are irreducible representations of G and the n | their multiplicities in the decomposition sum. It means that even in a cubic crystal the J-tensor should be ani-sotropic, so that off-diagonal elements will occur in the matrix partly removing the degeneracy in the cluster multiplets. For Si = S2 = % and the I-tensor being isotropic, axial and diagonal respectively, the energy levels are given in Fig. IV.1.
diagonal isotropéc o,*-J"*»! Fig. IV.la -JL(11-1>*|11» , ( | I - 1 > - | I 1 » -|10> -|00>
no interaction isotropic axial
Fig. IV.lb
Fig. IV.la. Energy levels and spin functions for S | axial and diagonal.
b. Numerical data for Cobalt in KMgF3.
S2 = V^ and for the J tensors being isotropic,
In this thesis, we shall be concerned with systems, where the zero field sphtting is either equal to zero (Co^ and Ni^ in MgO, Co^ in LiMgP04) or so small, that its effect on the magnetic properties is negligible (Mn^* in MgO and LiMgP04). In our discussion of the magnetic properties of coupled spin systems we shall, therefore, simplify our Hamiltonian by omitting the zero field splitting tensors.
For ions with no orbital degeneracy in their ground state, the anisotropic con-tributions to the exchange interaction usually are so small, that their effect on the magnetic susceptibility is negligible. For ions with an orbitally degenerate ground state, higher anisotropic contributions are to be expected. However, recent quanti-tative calculations performed by Levy et al. [3] have shown, that even for these ions the anisotropic parts are small compared to the isotropic part (see Fig. IV.lb). Our second simplification will therefore be the assumption of an isotropic exchange ten-sor. In this case we have:
where
J = Va trace ƒ , (4.7) so that Eq. (4.3) is now written as:
3(; = J S i - S a + H G i S i + H G 2 S 2 (4.8)
4.3.2 Energy levels and g tensors of cluster multiplets
We shall rewrite Eq, (4.8) in the more convenient form used by Griffith [I ] :
5Q = 2J Si. S2+ H^ E (g(') S[»)+ g(') S(2))hj = ?(;, + 3C H/3 (4.9) j,k •'
where we have taken the magnetic field as
H = (hx,hy,h,)H (4.10) and where j and k are subscripts for x, y, z.
In the absence of a magnetic field the matrix of 3Q on the basis of the IS, M) functions is diagonal because of the relation:
3(;) = 2 I S i - S 2 = J ( S ^ - S ? - S l ) (4.11) ( Is, M > functions are eigenfunctions of S^,S1 andS^).
From the eigenvalues of KQ.
E(S) = J [S(S+1)-S,(S,+I)-S2(S2+I)] (4.12) It follows that states IS, M ) belonging to a same cluster multiplet S are degenerate
in the absence of a magnetic field. Furthermore it is seen that the energy difference between two succeeding cluster multiplets is equal to:
E(S) - E(S-1) = 2 JS (4.13) The degeneracy is removed by applying a magnetic field. We shall derive a
general formula for the g-tensor of a cluster multiplet, starting from the single ion values.
First we consider the matrix elements involving the z-components of the spins:
<S, M \K Is, M') = 2 (g}J>< S, MIS("I S, M') + ^jX S, Ml S^^^IS, M'> )hj
30
and compare them to the elements:
<S,M I 2 gj.Szhj IS,M > = M 6 M M ' 2 gjzhj (4.15)
i i
where g is the g-tensor of multiplet S. It should be noted, that only the matrix ele-ments of Jf^ , which occur (on the diagonal) in a diagonal block give a contribution to the Zeeman splitting. The matrix elements of ^ in the off-diagonal blocks cause an equal shift in all levels arising from a same multiplet and only give a contribution to the temperature independent paramagnetism of the magnetic susceptibility. They will be discussed later. It should be noted that a 'diagonal block' in our matrix is formed by matrix elements between IS, M) states within one particular cluster mul-tiplet.
The element <S,M IS^'^ IS,M> = li is equal to: Ii = 2 [c(SiMiS2M2SM)]^Mi
M l , M 2
= - ( S i + l ) 2 (Si+Mi+l)[c(SiMiS2M2SM)]2 (4.16) M i , M 2
It is convenient to calculate Ii for M = S. From Wigner's formula we obtain: (Si+S2+S+2)(Si-S2+S+l)
li = ~^—^ ^—!— - ( S i + I ) 2(S+1)
^ (Si+l)Si-S2(S2+l)+S(S+l) 2(S+1)
Simüarly we have for <S, S Is^/^ Is, S> = I2 _ S2(S2-H)-Si(Si+l)+S(S-H)
^' - ~ 2(STÏ) ^^-'^^
Substituting (4.17) and (4.18) in (4.14) and putting (4.14) equal to (4.15) we find:
giz(S) = ' < ) < ) ) + '^i'>-4^>) ' ' ^ ' ' ' ^ s t o " ^ ' ' ^ (4.19)
We can obtain similar expressions for gjx and gjy by considering the matrix elements of Wx and iK^. With respect to (4.19) we can make the following notes:
a) The g-tensor in a coupled spin system has the same form as the formula for the Lande-g-factor for an atom in Russell-Saunders (L-S) coupling if we put g(S) = 2 andg(L) = l:
3^S(S-H)-UL+1) .420) 2 2I(J+1)
b) From (4.19) it follows immediately, that, if two identical ions are coupled, the g-tensors of each multiplet of the coupled system are equal to the g-tensor of the single ion.
4.3.3 The magnetic susceptibility
The magnetic susceptibility can be calculated by expanding the energy level of each state in a power series of the appUed magnetic field. If our perturbation Hamil-tonian is 3Cz, we have the following expansion:
E ( S , M ) = E ( S ) + P H < S , M I K I S , M > + ^ 2 H 2 2 (<S.Ml?C;is',M'y ^^^i) S'M' E(S)-B(S') If we include the x and y components in the perturbation Hamiltonian
( in = aC^ + oVx •*• "^y )
then in each diagonal block off-diagonal elements will occur. According to second order perturbation theory for degenerate states we must now expand the energy of state i as follows:
E(S,i) = E?(S) + E//3H + E?|32H2 (4.22) with: E? = an eigenvalue of the diagonal block S (the submatrix within multiplet S)
and :
E 2 - v ( i IJC' h ) ' E ? ( S ) - E 9 ( S ' )
where the summation is carried out over all states not belonging to multiplet S. It can be proved that E? has the same value for each state i arising from a same multiplet and that its value is equal to:
E 2 - v ( i l J C ' l i ) _ ^ <S,MI3CIS'.M') -.423^ ' i Ei(S)-Ej(S) S'M' E(S)-E(S')
According to Van Vleck the general formula for the magnetic susceptibility is given
"-" ' JI?(S)/.T (^•''"
iThe contribution of one multiplet S to the total susceptibility is:
Xs=N^M S [(Ej)2/kT-2Ef] ]]/(2S+I) (4.25)
32
where the summation is carried out over all states i from multiplet S. Hence the total susceptibility can shortly be written as:
' 2 ' ^ ' (2S+I)Xse-^(^>/"
S = S i - S 7
k ' (2S+1) 6-^(8)/"^
8=81-82
(4.26)
If a closed expUcit expression for Xs can be derived, then Xn, can readily be calcula-ted. It has been proved by Griffith, that the sum 2 (Ej f can generally be written
as: '
2 {E\f = i S(S+1) (2S+I) 2 ( 2 gjkhj)2 (4.27)
i 3 k j
If we have a random orientation of our molecules we must average over the angle of the magnetic field relative to axes of the molecule. Using ^(hjhi)> = '^ Sji, it is im-mediately seen,that
< 2 ( 2 g j k h j ) 2 > = | 2 2gf^ (4.28)
k j J k j
Neglecting the terms E? the average one-multiplet susceptibility can be written as
^Xs> = l ^ S ( S + l ) £ 2 g ? k (4.29) 9kT k i
In most cases the sum in this expression can be written as
ff8i'k=(g2^+gj+g^) (4.30) so that the multiplet susceptibility has the famiUar expression:
X s = ^ ( g ^ g ' y + g ^ z ) S ( S + l ) (4.31) (In this notation gxx is shortly written as gx.)
Since the g-factors in this expression can be calculated using Eq. (4.19), Xs is readily known. The contribution of each multiplet in the coupled spin system being known together with the multiplet energies E(S) (see Eq. (4.12)), the total susceptibility is immediately calculated with Eq. (4.26).
It can be proved, that for a cluster of identical ions the temperature indepen-dent paramagnetism is equal to the sum of the temperature indepenindepen-dent terms of the individual ions. However, Griffith has shown that an extra temperature independent term may arise from the interaction in a cluster with different ions. The extra term per multiplet is written as [ 1J:
2 E p ) = ^ (1-a^) 2 (g},;) - g[,^))^ (4.32) i(S) 36J jk
where
^ _ S i ( S i + I ) - S 2 ( S 2 + I ) S(S+1)
Putting our calculated values for the sums 2 (Elf and 2 E?
i(S) i(S)
in the formula for the one-multiplet-susceptibiUty (4.25), we obtain:
^ = N^2 [ S g ± a 2 (gj,)^ - N ê ! £ k ^ 2 (gO )_g|2))2 ] (4.33)
9 K 1 jk JOJ jk
With respect to (4.33) we can make the following notes:
a) The temperature independent term in Xs vanishes if the coupled system exists of identical ions.
b) The formula for Xs holds true if perturbation theory is vaUd, i.e., if J > ^H. 4.4 Three-spin systems
We shall now discuss the case, in which three spins Si, S2 and S3 are coupled. First we investigate, what type of multiplets there are to be expected and how many times they occur. This may be done by first coupUng the first spin Si to the second S2, which gives S' = Si+S2, , ISi—S2 I. Then couple each S'to S3 to give S = S'+Sa, , I s ' - S j I. For example for three Ni^^ions we have S' = 2, 1,0 and then S = 3 , 2 , 1 , 2 , 1,0, l.Thismay be written succinctly as S = 3, 2^, 1^,0. In case Si = S2 = S3 we have the following total spins:
S, = l / 2 : S = ( 3 / 2 ) , 1/2^ S i = l :S = 3 , 2 ^ l ' , 0
Si = 3/2 : S = (9/2), (7/2)^, (5/2)^, (3/2)r 1/2^ S i = 2 :S = 6 , 5 ^ 4 ^ , 3 ' * , 2 ^ 1 ^ 0
S, =5/2 :S = (15/2),(13/2)2,(Il/2)^(9/2)^ ( 7 / 2 ) ^ ( 5 / 2 ) ^ ( 3 / 2 ) ^ 1/2^ and any other case is easily worked out.
Now suppose the interaction Hamiltonian is:
3C = 2Ji 2S, •S2+212382-S3+2J3iS3. Si (4.34) Write S = S1+S2 and S = S1+S2+S3. Then from a theoretical standpoint we have
34
3C=2J2 S i S j = J S 2 - J S ? - I S l - J S | (4.35) from which it follows that
E(S) = J [S(S+I)-Si(Si+I)-S2(S2+I)-S3(S3+l)] (4.36) Thus the energy depends only on S and relative energies still satisfy the Lande'
inter-val rule given in Eq. (4.13).
The second case is when two of the coupUng constants are the same, say J2 3 = J31, but not the third. Here again we are fortunate in that we may write the Hamiltonian in the form:
3f=2Ji2SiS2 +2123(81+82)83 =2Ji2Si-S2+2J23S'-S3 (4.37) whence it follows that
E(S) = J , 2 [ S ' ( S ' + l ) - S i ( S i + l ) - S 2 ( S 2 + l ) ] + J 2 3 [ S ( S + l ) - S ' ( S ' + l )
-83(83+1)] (4.38) This is also very simple, although now E(S) depends on S as well as on S.
The third case is when all the coupling constants are unequal. There is no longer any general elementary method for obtaining the energies, although in any particular case the complete matrix of JCcan be calculated by actually writing out the coupled states and determining the matrix elements by elementary operations with angular momenta. In this thesis the third case will not be encountered. 4.S The biquadratic exchange term
The scalar product interaction operator 81-82 is by no means the only possi-ble one which commutes with the total spin S. In particular the somewhat more general operator
3Ce= 2J(S,-S2)-2j(S, 82)^ (4.40) also has this property. In a study on pairs of Mn^-ions, interstitially in MgO, Harris
and Owen [5] measured the positions of all levels except that having S = 5 and found the intervals did not approximate at all well to the interval rule shown in Eq. (4.13). However, they obtained satisfactory agreement with experiment by using the more general interaction given in Eq. (4.40). Its effects are shown in Table lV-1. Note that in this case, where the ratio j/J was found to be equal to 0.05, the second order con-tribution to the calculated intervals is really quite large, being on average some 27% of the total.
Table IV-1 Eigenvalues of JQ S 0 1 2 3 4 5 E(S) 0 2J+ 33j 6J+ 87j I2J+I38J 20J+I50J 30J+ 75j E(S) forj/J = 0 3.65 J I0.35J 18.90J 27.501 33.75J
In Chapter VI possible effects of the biquadratic exchange term will be taken into account.
References
1. J.S. Griffith, Stnicture and Bonding 10,87 (1972).
2. HP. Bakes, J.-F. Moser and F.K. KneubiJhl, J. Phys. Chem. SoUds 28,2635 (1967). 3. P.M. Levy and G.M. Copland, Phys. Rev. B 1,3043 (1970).
4. J.H. van Vleck, 'The Theory of Electric and Magnetic SusceptibiUties', Oxford (1932).
36
CHAPTER V
BIVALENT NICKEL IN MgO AND LiMgP04
5.1 Theory of the ground state of octahedral Ni 5.1.1 The spectroscopic ground state in the crystal field
The energies and wave functions belonging to the states of d" ions can be cal-culated by diagonalization of the Hamiltonian
?£'=JCo+JCe + 3{;+V (5.1) within a given d" configuration [1]. In this Hamiltonian JCo is the sum of the
one-electron potential energy and kinetic energy, "K^ is the one-electronic repulsion energy,
"K^ is the spin-orbit coupling energy and V is the Ugand field energy, representing the
electrostatic interaction between the partly fiUed shells and the rest of the complex ion.
Though the final results will be the same in each case, the actual detailed course of the calculation will depend on the order in which JJCg, "K^ and V are diagonaUzed. In this chapter we shaU perform our calculations on bivalent nickel using the strong field coupling scheme. In this scheme the one-electron orbitals are linear combina-tions of d-orbitals, forming the basis of irreducible representacombina-tions of the symmetry group of V. Thus in an octahedral field the d-orbitals are separated into two sets, e and t2, with the energies 3/5 A and —2/5 A respectively [ I ] .
If we take the determinantal wave functions having m electrons in t2g orbitals and n electrons in Cg orbitals as basis functions, the crystal field operator will be diagonal within these basic states. The set of states belonging to tfg eg is caUed a strong field configuration.
Upon diagonalizing JCg within the strong-field configurations, each configura-tion spUts into the so called strong field terms 2S+1 p^ where S is the spin quantum number and F an irreducible representation of the cubic point group 0. Considering in the case of octahedrally coordinated Ni^ the d' system as a d^ hole system we have:
e^ —> 3 A 2 + ' A I + ' E (5.2)
tigCg > - ^ T I + ^ T 2 + * T , + ' T 2 (5.3) t|g >• ^ T , + ' A I + ' E + ' T 2 (5.4)
The energy levels of octahedrally coordinated Ni**" are obtained by diagonalizing the matrix of <K^ and V on the basis of these strong field terms.
Non-vanishing matrix elements of Jfg only occur between strong field terms with the same F and S, while matrix elements of V only occur on the diagonal.
From (5.2). (5.3) and (5.4) it is immediately seen that the matrix breaks down in 3 (Ixl) blocks and 4 (2x2) blocks. In Table V-I a part of the matrix is shown.
Table V-I
The matrix of JCe and V on the basis of the strong-field terms for octahedrally coor-dinated Ni^; EQ = 28A + 50B + 2IC.
5fe+V
'4:
'T'(t2e) ' T i ( t i ) ^A2 Eo ^T2 Eo+A 'Ti(t2e) Eo+I2B+A 6B ' T i ( t l ) 6B Eo+3B+2AThe resulting energies are plotted as a function of the crystal field parameter A in Fig. V.I. In this diagrma ^Tj * and ^Ti'' are linear combinations of ^Ti (t2e) and ' T I (t2). It is seen that the ground state is a ^A2 level.
By spin-orbit coupUng two terms, 'T2(t2e) and ^T2(t2e), are mixed into this ^A2 term; in a first approximation the new ground state, belonging to the represen-tation T2 of the double group 0*, becomes:
lT24>= 'A2(e^T2i) - ^ 'T2(t2e,T2i) - -—f— •T2(t2e,T2i) (5.5) A A+8B+2C where Z is the spin-orbit coupUng constant.
The spectroscopic splitting factor is now calculated as: gx=gy=gz = < T 2 , l l L , + 2 S j T 2 , l > = 2 + ^
A Using Z = 649 cm'' and A = 8600 cm'' we get
g = 2.3I
which is somewhat higher than the experimental g value of 2.227 for Ni^'*' in MgO. In our formulae for the calculated susceptibilities we shall use the experimental (e.p.r.) values throughout.
If the Ni^Mon is surrounded by a distorted octahedron, in the Hamiltonian low symmetry crystal field operators should be introduced. In that case the three-fold degeneracy of the ground state will (partly) be removed.
38
^ A(cm ') Fig. V.1. Energy as a function of crystal-field parameter A for the d^ system Ni". Energies are in
cm"' and the ' S of the free ion is not shown. 5.7.2 The spin Hamiltonian
The ground state of octahedrally coordinated Ni^ can also be described in terms of the spin Hamiltonian [3 ]:
on the basis of fictitious spins Is, M). The parameters g, D and E can be expressed in terms of the spin-orbit coupling constant and the crystal field parameters. They may be determined experimentally by e.p.r. measurements. For Ni^ in MgO, D and E are both equal to zero. In LiMgP04 Ni^* is surrounded by a distorted octahedron of oxygen ions; in that case, due to the effect of D and E, the degeneracy in the ground state wih be removed.
S.2 Formulae for cluster susceptibiUties
The isotropic part of the exchange Hamiltonian for the 'A' type binuclear cluster is written as:
3 C = 2 J A S I - 8 2 (5.8)
The eigenvectors of W are basis functions of the irreducible representations D°, D ' and D^ of the direct product group D ' iS D ' :
D' ® D ' = D° + D ' + D^
The energies of the resulting singlet, triplet and quintet states are (see Eq. (4.12))
E(J) = JA [S(S+l)-Si(S,+l)-82(82+1)] , (4.12) with Si and S2 being the spin quantum numbers of single ions and S being the
quan-tum number of the cluster spin. Since, in our case. Si = S2 = 1, (4.12) reduces to
E(J) = JA [ 8 ( S + l ) - 4 ] (5.9) with S equal to 0,1 and 2 for the cluster singlet, triplet and quintet, respectively.
Since there is no orbital degeneracy in the ground state ^A2, the interaction takes place between 'true' spins S = 1.
The magnetic susceptibility can be calculated by extending (5.8) with the Zeeman term (see Eq. (4.9)). Assuming an isotropic g-tensor, for cluster 'BINA' the eigenvalues of the spin Hamiltonian are:
E ( S , M ) = J A [ S ( S + 1 ) - 4 ] +g^HM (5.10) withM = S , S - l , , - 8 .
Thus, the g-values of the single ion triplet and cluster multiplets are equal, which is a generally observed phenomenon in e.p.r. measurements and which can be predic-ted by theoretical considerations (see Chapter IV).
40
susceptibihty of cluster 'BINA' has been derived:
, (BINA) = N ê i [g^ (2e^^A/.^T , I O , - 6 U / K T , ^ ^^ ^^^ ^^ '^'T 1 +3e"2^A/kT+5g-6JA/kT
Following the same vector coupling scheme, the lower states of trinuclear clusters are found to consist of one singlet, three triplets, two quintets and a septet:
D ' iS D ' ® D ' = 0 " + 3 D ' +2D^ +D^.
The spin hamiltonians and their eigenvalues for all clustertypes are Usted in Table V-2. Energy spectra of the single ion and the clusters 'BINA', 'T-AB' and 'BINB' are shown in Fig. V.2.
Table V-2
Spin hamiltonians and eigenvalues for different types of clusters (isotropic exchange) name 8 BINA BINB TAAA TAA TABA TBB spin hamiltonian 2JASi-S2+g0H.(Si+S2) 2JBSi-S2+g|3H.(Si+S2) 2 J A ( S I - 8 2 + S 2 - S 3 + S I - S 3 ) + +ï/3H<8i+S2+S3) 2 J A ( S I - S 2 + S 2 - 8 3 ) + +g^H<Si+S2+S3) 2 J A ( S , - S 2 + S , - S 3 ) + 2 J B 8 2 - S 3 +g/3H<8i+82+83) 2 J B ( S I •82+82-83)+ +ö3H.(Si+82+S3) eigenvalues E(J) J A [ J ( J + 1 ) - S I ( S I + 1 ) - S 2 ( S 2 + 1 ) ] + « ^ M J H J B [ J ( J + I ) - S , ( S I + I ) - S 2 ( 8 2 + 1 ) ] + B ^ M , H J A [ J ( J + 1 ) - 3 8 I ( S I + 1 ) ] + S ^ M J H JA [J(J+I)-k(k+I)-S,(S,+l)]+g^MjH JB[k(k+l)-2S,(Si+l)] + +JA [J(J+l)-k(k+I)-Si(Si+I)]+«pM,H JB[J(J+I)-k(k+l)-Si(S,+l)]+g^MjH Si - S2 ~ S3 = I
k = 0 , 1 , 2 = S' (see Chapter IV) l k - S , l < J < k + S ,
- J < Mj < J.
J (= 'total spin') = spin quantum number of cluster multiplets = S
The magnetic susceptibility of each trinuclear cluster has been calculated from spin hamiltonian eigenvalues as done for the binuclear clusters. For tetranuclear clusters no detailed calculations have been made. Their susceptibility is taken as 4/3 times
2JA O--iJA"' E(J)
•
S BINA BINB TAB
Fig. V.2. Energy spectra of the single ion 'S' and the clusters 'BINA', 'BINB' and 'TAB'.
5.3 Experimental
The magnetic susceptibility measurements on Ni doped MgO powders were per-formed in the temperature range 5 to 300 K using a P.A.R. vibrating sample magneto-meter, type 155, in combination with a caUbrated magnet of a Varian V-4500 spec-trometer. The magnetometer was calibrated by measuring the saturation magnetiza-tion of a standard Ni sample, deUvered by P.A.R.. Measurements were performed on four different samples NixMgi _xO, x ranging from 0.0068 to 0.0465. Diamagnetic corrections were carried out by using a value of -0.28-10"^ c.g.s. for the susceptihi-Uty per gram MgO (Xm = - 1 1 -lO"* c.g.s.). This value was determined by direct meas-urements on the undoped substance.
Homogeneous NixMgi _xO samples were prepared as follows:
From a homogeneous NiCl2—Mga2 solution the Ni^ and Mg^ ions were precipita-ted together by sodium carbonate. The hydroxycarbonates formed were converprecipita-ted into very pure homogeneous magnesites by a hydrothermal treatment (I50°C, 50 atm. CO2 pressure) as described by E. Doesburg [2]. The decomposition of the magnesites into the oxydes was carried out at 1000°C in nitrogen atmosphere. In the e.p.r. spectra of these samples no Ni'**" or Ni'' signals have been observed [3], which
0
^2JB
42
is an indication that the concentration of these ions is less than 0.001%. The concen-trations X were determined by electrolysis and supplementary titrimetric analysis of a solution of the end product in HCl, the relative accuracy being better than 0.5% for samples with high concentrations and better than 1% for the sample with the lowest concentration.
Homogeneous LiMgi _xNixP04 samples were prepared by spraying a concen-trated solution of weighted amounts of LiQ, (I—x) MgCl2-6H2 0, x NiCl2-6H20 and (NH4)2HP04 (aU p.a.) in HCL containing water, in a 500°C quartz vessel. The result-ting powder was heated for six hours in nitrogen atmosphere at 800°C. LiMgi -xNixP04 was formed according to the reaction:
Lia+(l-x)Mga2+xNia2+(NH4)2HP04 ^ UMgi_xNixP04+HClt+2NH4Clt. X-ray photographs showed the orthorhombic olivine structure. An analysis of Ni in the end product showed, that the concentrations x did not change during the process. 5.4 Results and discussion
5.4.1 Ni^ in MgO
Typical examples of the experimental data are given in Fig. V.3. In this graph the product of the molar susceptibility and temperature for the samples Ni 0.0178 Mgo.98220 and Nio.0465 Mgo.953sO are plotted against temperature. The single ion susceptibihty was determined very accurately from e.p.r. data.
X s = ^ g ' 8 ( S + l ) + a (5.12) 3kT
with N = Avogadro's number ^ = Bohr magneton g =2.227
8 =1
a = temperature independent paramagnetism per mole Ni^
= 0.0002 (calculated using BaUiausen's formula [4] with A = 8.600 cm''). The measurements clearly show a deviation from the single ion susceptibility, which is typically temperature dependent and closely related to the concentration. We pro-pose to interprete the experimental data in terms of isolated spin clusters. In this model the total susceptibihty is written as a stun over contributions from the single ions (S) and aU clusters i:
Xtotai = Ps(x)Xs+2Pi(x)xi . (5.13)
i
From (5.13) it is clear, that two experimental procedures can be followed to deter-mine the exchange coupling constants JA and J B .
2.00 180 160 1.40 UO
oso
OMQ 020 XT(c.g.s.°K) — single lop "^ N'0.0178^90.9822° -^ N'0.0i65^90 9535° •Temp.(V) 30 60 90 120 150 180 210 2i0 270 300 Fig. V.3. Molar magnetic susceptibilities of Ni^* in Ni^ ; „ , ^ , Mg„ , j , 5 U multiplied by thetemp-erature. The solid curves are the calculated single ion susceptibihty and the least squares fits to the experimental points.
a) Via a Xt(x) vs. x plot
For a fixed temperature T, Xt is plotted as a function of x in the series of sam-ples Mgi_xNixO. These curves are linear at small concentrations x (see Table 1II.2). At higher concentrations a deviation occurs caused by the increasing number of tri-nuclear clusters. Xs is determined by extrapolating to infinite dilution, while the sum XB1NA+ XBINB is derived from the slope at x = 0. By plotting the sum obtained at a number of temperatures as a function of temperature, x BINA(T) and XBINB(T) can be solved separately if there is enough difference between JA and J B . In a computer-ized least squares fit procedure a correction for trinuclear clusters can be included. b) Via Xt vs. T plots
Since xt shows characteristic temperature dependent deviations from xs> caus-ed by the interactions 'A' and 'B' the set parameters (JA, J B ) can be determincaus-ed by iterating to obtain a least squares fit of calculated and experimental data.
44
clearly is the most efficient one. It should be noted, that the more JA and JB differ, the better their effects in the observed curve are separated and the more accurate the determination of these parameters is.
We have interpreted our experimental data following method b). Since the error in the calculated value of Xt, introduced by making an estimate for tetranuclear cluster susceptibilities, might exceed 0.3% for x > 0.02, in a first analysis only sam-ples in the lower concentration range (x = 0.0068 to O.OI 78) have been regarded. Using Table III-2, the calculated total susceptibihty (5.13) for sample
Nio.0068 Mgo.9932 0 can be written as:
Xtotal(0.0068) = 0.8842 xs + 0.0683 x BINA/2 + 0.0332 x BINB/2 + (5.14) + 0.0017 X TBB/3 + 0.0009 X T A A A / 3 + 0.0035 XTAA/3 + + -.0014 XTABA + 0.0054 XTAB/3 + 0.0019 XTETRA/4 For Xtotal (0.0178) a similar calculation has been made.
Xs being known, Xt only includes the parameters JA and J B . A least squares fit of (5.13) to the experimental plots results in the foUowing values for JA and JB:
X = 0.0068 : JA/k = 2.32 K (Je/k = 146.8 K) X = 0.0178 : JA/k = 2.45 K (JB/k = 199.0 K)
The quality of the obtained fit can be judged from Table V-3 and Fig. V.3. As a consequence of the decreasing accuracy in the measurements on small concen-tration samples in the higher temperature range, temperature has not been raised above 70 K and 90 K respectively. In this range the JA parameter can be deter-mined with sufficient accuracy. Since the effect of JB (± 150 K) manifests itself in the range 70 to 500 K (Fig. V.4) an accurate determination of JB requires high temperature measurements on samples with a higher concentration of Ni ions. It follows from Table V-4 that in the range > 70 K the complete ensemble of clusters behaves as if just one superexchange path, being the 180° type, was present. Apparent ly, all ions coupled through a 90 superexchange path can now be regarded as single ions so that type of clusters and their concentration can be taken identical to those to be expected for a simple perovskite type structure. As can be seen in Table III-3, the number of tetranuclear clusters in this simpUfied interaction pattern at x = 0.04 is less than the number in the original lattice at x = 0.02. It means that in a special high temperature analysis, which is required for a more accurate determination of J B , the use of the experimental data on samples with x = 0.0312 and 0.0465 is justified.
Using Table II1-3, the calculated susceptibihty of Nio.0312 Mg 0.9688 0 can be written as:
Xt(0.03I2) = 0.8253-xs + 0 . 1 3 7 2 X B I N B / 2 + 0.0295x TBB/3 (5.15) A similar expression has been derived for Xt(0.0465).
Using again a least squares fitting procedure JB (and J A ) are now found to be: X = 0.0312 : Je/k = 152.0 K (JA/k = 2.0 K)
X = 0.0465 : JB/k = 146.4 K (JA/k = 2.2 K)
In Table V-5 experimental and calculated susceptibihty values are compared. It should be noted that as a further refinement to our simphfied interaction model the effect of homogeneous 'A' type clusters ('BINA', 'TAA' and 'TAAA') has been taken into accoimt. (In our first approximation of Xt in the high temperature range XBINA/2, XTAA/3 and XTAAA/3 were assumed to have the same value as Xs. see Fig. V.4). After this correction the calculated susceptibility at sufficiently high temperature is practically identical to the susceptibihty of the real system.
Table V-3
NixMgi _xO-samples with low Ni-concentrations; determination of J^ x = 0.0068 (JA/k = 2.32)
Temp Xobs Xs Xcaic K cgs cgs cgs 5.14 0.2202 0.2414 0.2218 5.70 0.1992 0.2177 0.2008 5.95 0.1913 0.2086 0.1926 6.44 0.1792 0.1927 0.1784 7.17 0.I6I5 0.1713 0.1608 8.72 0.1335 0.1424 0.1330 10.70 0.1102 0.1161 0.1090 13.70 0.0861 0.0907 0.0856 17.73 0.0665 0.0701 0.0665 22.79 0.0515 0.0546 0.0519 27.10 0.0435 0.0460 0.0435 32.10 0.0368 0.0388 0.0369 36.26 0.0331 0.0344 0.0327 39.80 0.0299 0.0314 0.0299 45.80 0.0261 0.0273 0.0261 52.66 0.0226 0.0237 0.0228 58.07 0.0207 0.0216 0.0207 63.96 0.0188 0.0196 0.0188 70.21 0.0171 0.0177 0.0172 x = 0.0178 (JA/k = 2.45)
Temp Xobs Xs Xcaic K cgs cgs cgs 5.14 0.I9I4 0.2414 0.1913 5.69 0.1724 0.2181 0.1752 7.20 0.1413 0.1724 0.1414 8.79 0.II77 0.1413 0.1177 10.72 0.0976 O.I 159 0.0979 13.63 0.0777 0.0912 0.0781 17.63 0.0616 0.0705 0.0611 22.68 0.0483 0.0549 0.0480 27.10 0.0408 0.0460 0.0404 31.90 0.0351 0.0391 0.0345 39.47 0.0281 0.0316 0.0280 45.80 0.0248 0.0273 0.0242 52.66 0.0213 0.0237 0.0211 58.00 0.0193 0.0216 0.0192 63.96 0.0175 0.0196 0.0175 70.20 0.0158 0.0179 0.0159 77.00 0.0145 0.0163 0.0146 84.40 0.0133 0.0149 0.0133 91.75 0.0123 0.0137 0.0123
30 'SO 90 120 150 180 210 240 270 300 Fig. V.4. Reduced molar cluster susceptibilities x(BINA)/2 and x(BINB)/2 multiplied by the
temperature, given as derived from the experimental data on the samples
Ni„-„i 7 8^^ëo.9 8 2 jO^""* Nio„4jsMg„_, 53 5O respectively. The solid curves are the calculated susceptibilities for the single ion and for the clusters 'BINA' (.ifj^ = 2.45 K) and 'BINB' (Jg/k = 146.4 K).
Table V-4
Calculated total susceptibiUty Xt of NixMgi _ xO per mole Ni^"^ for i\lk = 2.5 K and JB/k=150K.
a) taking into account all types of clusters in Table III-2: Xi b) taking into account only the 180° interaction: X2
x) as b) with a correction for the clusters BINA, TAA, TAAA: Xs-x = O.OI T K 10 30 50 70 100 130 160 200 Xl c.g.s. K 1.122 1.151 1.157 1.160 1.163 1.166 1.170 1.174 X2 c.g.s. K 1.166 1.166 1.166 1.166 1.167 1.170 1.172 1.186 X3 c.g.s. K 1.126 1.153 1.158 1.160 1.164 1.167 1.170 1.174 X = 0.02 Xl c.g.s. K 1.012 1.059 1.069 1.074 1.079 1.085 1.090 1.097 X2 c.g.s. K 1.084 1.084 1.084 1.085 1.087 1.191 1.097 1.101 X3 c.g.s. K 1.024 1.064 1.072 1.076 1.081 1.086 1.093 1.098
Table V-5
NixMgi-xO-samples with high Ni-concentrations; determination of JB x = 0.0312 (JB/k =152.0)
Temp Xobs Xs Xcaic K cgs cgs cgs 5.14 0.1729 0.2414 0.1767 5.69 0.1606 0.2I8I 0.1615 6.39 0.1453 0.1942 0.1456 7.20 0.1318 0.1724 0.1308 8.73 0.1109 0.1422 0.1097 10.72 0.0913 0.1159 0.0908 13.67 0.0730 0.0909 0.0723 17.67 0.0569 0.0704 0.0566 22.68 0.0449 0.0549 0.0445 27.10 0.0376 0.0460 0.0375 31.90 0.0322 0.0391 0.0321 45.29 0.0229 0.0276 0.0228 57.40 0.0180 0.0218 0.0180 70.01 0.0148 0.0179 0.0149 83.83 0.0124 0.0150 0.0125 91.75 0.0113 0.0137 0.0115 106.70 0.0099 0.0118 0.0099 123.60 0.0086 0.0102 0.0087 141.50 0.0076 0.0090 0.0076 293.30 0.0040 0.0044 0.0039 300.00 0.0038 0.0043 0.0038 x = 0.0465 (JB/k= 146.4)
Temp Xobs Xs Xcaic K cgs cgs cgs 7.20 0.1166 0.1724 O.I 169 8.73 0.0987 0.1422 0.0988 10.69 0.0830 0.1162 0.0825 17.62 0.0525 0.0706 0.0521 31.90 0.0298 0.0391 0.0296 45.70 0.0209 0.0273 0.0209 52.70 0.0181 0.0237 0.0182 58.00 0.0165 0.0216 0.0166 63.96 0.0150 0.0196 0.0151 70.00 0.0137 0.0179 0.0139 77.00 0.0126 0.0163 0.0127 84.00 0.0116 0.0150 0.0117 90.80 0.0108 0.0139 0.0108 105.80 0.0093 O.OI 19 0.0094 121.00 0.0082 0.0104 0.0083 139.00 0.0073 0.0091 0.0073 155.00 0.0067 0.0082 0.0066 169.50 0.0061 0.0075 0.0061 181.50 0.0058 0.0070 0.0057 293.00 0.0037 0.0044 0.0038 296.00 0.0037 0.0044 0.0037
In order to illustrate the contribution of clusters to the total susceptibiUty, in Fig. V.4 XBINA/2 is given as derived from the x = 0.0178 curve assuming JB/k to be 150 K and x BINB/2 is given as derived from the x = 0.0465 curve assuming JA/k to be 2.3 K.
Considering the data on low concentration samples as decisive for JA and the data on samples with higher concentrations as decisive for J B , our conclusion is:
JA/k= 2.3 ± 0.2 K JB/k = 150± l O K .
48
5.4.2 Ni2->-in LiMgP04
Typical examples of the experimental data on the system liNixMgi _xP04 are given in Fig. V.5.
1.40-.T (e.9.s.K) 1.20 1.00 0.80 0.60 0.40 0.20 •single ion ^Li Mg Ni PO ^0 950 0.050 4 *jli O O i O — o o o o I o o o 30.00 60.00 90.00 120.00 TEMP. (K)
Fig. V.5. Molar magnetic susceptibilities ofNi" in LiMg,.,,,Ni,„„,PO, and LiMg„., j,Ni„ „ j„PO, multiplied by the temperature as a function of temperature. The solid curves are the least squares fits to the experimental points.
