Investigation of magnetic interactions in sulfides by means of magnetic resonance

INVESTIGATION OF MAGNETIC

INTERACTIONS IN SULFIDES BY MEANS

OF MAGNETIC RESONANCE

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INVESTIGATION OF MAGNETIC

INTERACTIONS IN SULFIDES BY MEANS

OF MAGNETIC RESONANCE

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. IR. F. J. KIEVITS,

VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 20 SEPTEMBER 1978 TE 14.00 UUR

DOOR

GERARD VAN VEEN

SCHEIKUNDIG INGENIEUR GEBOREN TE AMSTERDAM

A

OOC^

BIBLIOTHEEK TU Delft P 1788 6004

Dit proefschrift is goedgekeurd door de promoter

Prof.Dr.Ir.L.L.van Reijen

CONTENTS

I INTRODUCTION 1

II SIMULATION AND A N A L Y S I S OF EPR SPECTRA OF PARAMAGNETIC IONS 5

IN POWDERS

1 Introduction 5

2 Q u a l i t a t i v e interpretation o f p o w d e r spectra 1 0

3 Computation of the full powder spectrum 16

4 Interpolation scheme for the simulation of powder spectra 19

5 Application to the simulation of an actual spectrum 22

5 Appendix Computational details 24

5.1 Resonance position 2 4

5.2 Derivatives of the resonance position 2 5 6.3 Average transition probability 2 7

5.4 Approximation of S (H ) 28

'^ a o

III SPIN HAMILTONIAN FOR A PAIR OF INTERACTING PARAMAGNETIC IONS 3 0

1 Introduction 30

2 Formulation of the pair Hamiltonian in terms of irreducible 3 1

spherical tensor operators

3 Approximations for strong isotropic interactions 38

4 Applications 42

IV SELECTION, PREPARATION, AND CHARACTERIZATION OF SAMPLES 46

1 Survey of suitable systems 46

2 Experimental selection of systems for further study 5 1

3 Preparation and characterization of powder samples 5 3

3.1 General p r e p a r a t i o n technique 5 3

3.2 Spinel compounds 54 3.3 Chalcopyrite compounds 6 0

4 Growth o f single crystals by chemical vapour p h a s e transport 6 5

4.1 Introduction 65 4.2 Chemical vapour phase transport 67

4.3 Cdin S :Cr 69 4.4 Chalcopyrites 70

V SYSTEMATIC ANALYSIS OF THE TRANSPORT RATE OF CuAlS 74

VI MAGNETIC RESONANCE MEASUREMENTS 82 1 Superhyperfme interactions in some sulpho spinels 82

1.1 EPR measurements of Mn(II) and Fe(III) in ZnAl S and CdY S 82

1.2 ENDOR measurements of Cr(III) in CdIn S 85

1.3 Spin delocalization 87

2 Single ion epr spectra of Fe(III) in sulpho chalcopyrites 94

2.1 Results 94

2.2 Accuracy of results 98 2.3 Computational method 100

3 EPR spectra of Fe(IIl) pairs in CuAlS and CuGaS 102

3.1 Experimental observation 102

3.2 Prediction of spectra of pairs of strongly interacting ions 103 3.3 Assignment of experimental spectra to a specific pair 107

3.4 Evaluation of results obtained 112 4 Discussion of magnetic pair interactions m CuFeS and 3-MnS 115

4.1 Magnetic structure 115 4.2 Neel temperature 119 4.3 Curie-Weiss temperature 120 VII CONCLUSIONS 121 List of symbols 124 References 125 Summary 129 Samenvatting 131

Hierbij bedank ik alien, die aan de totstandkoming van dit

proefschrift hebben meegewerkt.

CHAPTER I

INTRODUCTION

Since the late thirties, the understanding of the electric

and magnetic properties of oxides of transition metals has

been greatly stimulated by their application in the electronics

industry. It was understood that the mutual substitution and

site preference m binary and ternary compounds are often

governed by ion size, and that their magnetic and electric

properties can be described m terms of localized magnetic

moments and electric charges, both corresponding with some

well defined valence states.

With the introduction of crystal field theory and the

simultaneous development of paramagnetic resonance also a

better understanding of the spectroscopic properties and other

factors governing site preference has grown.

A new rapid development occurred since 1960, characterized by

an increasing interest m compounds with S, Se, Te (chalcogenides)

and P, As, Sb (pnictides). Industrial application were derived

among other things from their opto-electric and opto-magnetic

properties. Many phenomena had to be explained by the band

theory of electric conductivity and electronic magnetism and

by models m which itinerant electrons affect the interactions

between localized magnetic moments. The flexibility of important

properties, like band gap, refraction index, and optical

absorption advance applicability of these compounds.

Of these various compounds, the sulfides are expected to be

most closely related to the oxides. In this context it is

striking, that a relatively small number of binary sulfides

has the same crystal structure as the corresponding oxides,

while also in the ternary sulfides new classes of structures

predominate. This phenomenon is thought mainly to be due to the

the oxygen anion. Thus the sulphur ions prefer lattice sites

with a large electric field gradient.

As regards the electric and magnetic properties, certain

transition metal ions, Cr, Mn, Fe and Ni, behave rather

similarly in oxides and sulfides and localized models suffice

for the description. For other transition metal ions the

Itinerant electron models predominate easier in sulfides

than m oxides. The behaviour of Cu m sulfides, usually

preferring valence state one and tetrahedral coordination,

shows strikingly little resemblance to its behaviour m

oxides.

The general assumption in explaining the differences m

properties between oxides and sulfides is that the covalency

m sulfides is much larger, i.e., that the molecular orbitals

show a stronger mixing of cation and anion orbitals. However,

there is little quantitative information on this effect,

neither from experiments nor from molecular orbital calculations.

Also It IS still an open point to what extent the increased

covalency will affect, for example, the magnetic properties.

The present study was undertaken with this problem in mind.

It IS generally assumed that the magnetic properties of

semi-conductors with localized magnetic moments can be

described by pairwise interactions between the magnetic moments.

However, the information from the magnetic properties (magnetic

susceptibility, magnetic structure and sublattice magnetization

as a function of temperature) is too little to obtain a

reliable description of the magnitudes of the various interactions;

usually It IS very difficult to obtain even two interaction

parameters. Hence, only in the case that one interaction

parameter is strongly dominant, one can expect adequate

information from the properties of the pure magnetic compound.

The present work was set up to explore this gap by utilising

systems where a small amount (of the order of 1%) of the

magnetic ions were incorporated in a diamagnetic host lattice.

The most direct way is the measurement of the paramagnetic

resonance (epr) of pairs of magnetic ions formed accidentally by random substitution. Especially orbital singlet ions are accessable to such measurements. Pairs of V ( I I ) , Cr(III) and Mn(II) have been measured in various oxide host lattices. The only measurement available in sulfides thus far are those of Mn(II) pairs m ZnS. We have investigated the suitability of ternary sulfides as host lattices for this type of measurements, with the special aim to be able to study the effect of variation of lattice distance and of the contribution of interaction paths over various diamagnetic cations. Two types of compounds have been considered m particular: normal spinels and chalcopyrites.

A second method to study the delocalization of electron spins is the measurement of the superhyperfme interaction between the electron spin of the substituted paramagnetic cation and the nuclear spins of the neighbouring diamagnetic cations. These interaction can be measured by means of epr or electron nuclear double resonance (endor).

In this thesis only a first step to this ambitious programme has been set. Much attention had to be paid to important preliminaries a) The analysis of fine-structure epr spectra of powder samples.

In many cases powder measurements form an essential

introduction to single crystal work. For spin 5/2 such an analysis is still quite complicated. Especially the

simulation of a spectrum with given parameters is a problem that requires further attention.

b) The formulation of a spin Hamiltonian for interacting Fe(III) pairs and the mathematical background required for the

analysis of the epr spectra of such pairs. The analysis of Fe(III) pairs requires higher order anisotropy terms than the systems studied up to now. Making a consequent use of the properties of "irreducible spherical tensors", various stages in the analysis of the epr spectra of the pairs are largely simplified:

reference coordinates.

- The relation between the parameters of the spin Hamiltonian in terms of the total spin of the pair and those of the spin Hamiltonian describing the local anisotropy and interactions m terms of single ion spins.

- The calculation of the matrix elements.

c) The growth of doped single crystals. The epr measurements of the interaction between paramagnetic ions as well as the endor measurements of the delocalization of the unpaired electrons over the surrounding diamagnetic cations turn out to be practically impossible without sigle crystals. d) Finally a systematic study had to be made of the feasibility

of the preparation of a coherent series of doped sulfides and the possibility of observing superhyperfme structures of isolated magnetic ions, and of identifying epr spectra of pairs of magnetic ions against the background of the much stronger single ion spectra.

It turned out, that up to now only one type of pair spectrum could be identified provisionally, whereas some measuerements of spin delocalization by means of epr and endor could be performed and compared with literature data.

In chapter II and III of this thesis the analysis of the epr spectra of powders and the formulation of the spin Hamiltonian for Fe(III) pairs will be presented. Chapter IV is concerned with the selection, preparation and characterization of the host lattice and the magnetic ions that are most suitable for the systematic study of the magnetic interactions. A systematic analysis of the crystal growth experiments with CuAlSp will be presented m chapter V. In chapter VI the magnetic resonance experiments and their interpretation are presented in detail. A discussion is given of the impact of the results for the understanding of the magnetic properties

CHAPTER II

SIMULATION AND ANALYSIS OF EPR SPECTRA OF PARAMAGNETIC IONS

IN POWDERS

II.1 INTRODUCTION

In many applications of epr spectroscopy, only spectra of

randomly oriented samples (powders or glasses) can be obtained.

Generally, such spectra can be obtained more simply than

single-crystal spectra. The interpretation of powder spectra,

however, poses an enormous problem when the effective spin is

high and the symmetry is low. In fact very few powder spectra

of S=5/2 systems have been analyzed. An important difficulty

IS the excessive computer time necessary to simulate the

spectrum for a particular guess of the parameters of the spin

Hamiltonian. For example, Pedersen and Toftlund (1) stated

that they needed 30 hr to compute only one spectrum of a

slightly orthorhombic S=3/2 system. In this chapter the

problems encountered m the analysis of a powder spectrum will

be analyzed and an adequate procedure will be presented for

the simulation of S=5/2 spectra.

In an epr experiment one looks at the absorption (Q) of

electromagnetic radiation with a fixed frequency (v) as a

function of the strength (H) of the magnetic field (H), which

has a fixed orientation relative to the sample. Experimentally

one usually measures 9Q/3H. In general, the spectrum will

depend on the orientation of the magnetic field relative to

the sample.

A powder spectrum is a superposition of spectra of single

crystals with different orientations relative to the magnetic

field. In the following it is assumed that there is random

orientation of the crystallites, i.e. that the orientations of

the crystallites relative to the magnetic field are

The values of the energy levels of the paramagnetic ion as a

function of the orientation and the strength of the magnetic

field are described by means of a "spin Hamiltonian" /'(H) and

a set of spin functions |s,m> (2). The quantity F contains a

number of adjustable parameters to be determined m such a way

that the eigenvalues of the matrix of h on the basis of the

spin functions are equal to the energy levels of the

paramagnetic ion.

Absorption of radiation is possible when the energy difference

of two levels is approximately equal to the energy quantum of

the radiation:

A (H) ' |E (H)-E (H)I ^ hv (II.1)

The transition probability m the external magnetic field

H+Hi exp (•£2TTvt) , in which Hj is the amplitude of the magnetic

vector of the radiation, can be written as

P^_|(H,H,) = M^j (H,Hi) -f (A^^ (H)-hv) [11.2]

with

M^_^ (H,Hi) = |<i| 1 (Hi) |3>|' (II.2a)

where //j exp (i27TVt) is the time-dependent part of the

Hamiltonian of the system m the external field, and

f(A (H)-hv) IS the lineshape function, indicating the degree

to which the resonance condition of Eq.(II.l) must be

fulfilled in order to provide absorption of microwave radiation.

The lineshape function must be normalized (as follows from the

time dependent perturbation theory):

f(x) dx = 1 (II.2b)

There are two types of functions often applied to describe the

lineshape, Gaussian and Lorentzian, respectively:

f(x) = r"^/2A exp{-2(x/r )'}

^ ^ (II.3)

I - 1

f(x) = r /2737 (i+|(x/ry)'

normally are rather insensitive to details of the lineshape function other than the quantity T , representing the distance between the inflexion points.

As mentioned earlier, m recording an epr spectrum the strength of the magnetic field is varied at constant microwave frequency and at constant orientation of the sample with respect to the magnetic field. This means that in Eq.(II.2) the lineshape function should be replaced by one of the type g(H-H ) , in which H IS the strength of the magnetic field at which the resonance condition is fulfilled exactly. The subscript a identifies one of the solutions of Eq.(II.l), obtained by varying the strength of the magnetic field in a fixed

direction. To each a there corresponds a pair of states i and 3 between which the transition takes place. To determine the peak-to-peak I m e w i d t h (the distance of the inflexion points) on the magnetic field scale, it is reasonable to use the approximation 3A A (H) = A (H ) + (H-H ) • W ^ ( H ) . a a a a 3 H ^ a Because A (H )=hv, « « 3A A (H) - hv = (H-H ) - ^ ( H ) . a a 3H a

Substituting this equation into one of the two functions of Eq.(II.3] gives the peak-to-oeak I m e w i d t h on the magnetic field scale,

3H

'"H,a ., I r , (II.4)

a

3hv V, a and Eq.(II.2) becomes

3H

P^(H,Hi) = M^(H,Hi)

a

g (H-H ) , (II.5)

a a

in which g (x) is the lineshape function on the magnetic field scale, having the same form as f(x) but containing f,, instead of r . Thus the integrated intensity of an experimental epr line ( m a fieldswept spectrum) is |3H /3hv| times the

frequency-and is illustrated in Fig.II.1.

Fig.II.1 Transition probability as a function of the magnetic field and the frequency, respectively.

Concerning Eq.(II.5), it should be noted that |3H /3hv| and

consequently g (H-H ) will depend on the orientation of the

magnetic field. To make the notation less complicated this

dependence will not be indicated explicitly.

Some remarks should be made with respect to Eq.(II.4}.

First, in dealing with a single line of a single paramagnetic

ion, it is reasonable to assume T

independent of a and

of the orientation and strength of the magnetic field.

Secondly, in practice the resonances are composed of groups

of closely spaced lines (inhomogeneous broadening) originating

from unresolved hyperfine splittings or from such sample

imperfections that different ions in the sample should be

described by slightly different spin Hamiltonian parameters.

In these cases one may use a "phenomenological" linewidth

that takes both types of inhomogeneous broadening into account;

r = r + y r

v,a o p P

3A.

3p

or alternatively,

3H

H, a o

_3hv

+

I

r

3p

(II.6:

Here p refers to the parameters in the spin Hamiltonian, and

r represents the standard deviation of the parameter p.

According to Eq.(II.5) the spectrum of a single crystallite IS obtained as

I P„(H,Hi) = I M^(H,H,)

a a 3H 3hv •g (H-H ) a a

As mentioned earlier, to obtain the powder spectrum one must superimpose the single crystal spectra for different orientations of H and H i .

In the usual experimental setup Hi is perpendicular to H. Hence both directions can be described by one set of three Euler angles relative to the Cartesian reference coordinates. The references axis are chosen most conveniently along the principle axis of the g tensor. The direction of H can be described by the first two Euler angles 0 and (|) , which are equal to the polar angles 6 and (|) as indicated in Fig. II. 2.

Fig. II. 2 The angles 9 and ([) describing the direction of the magnetic field with respect to the reference coordinates.

The direction of H can be described by the Euler angle the powder spectrum becomes

IT TT IT Q(H) = Thus

dx dtf) de sine-^M (H,Hi )

•' •' a o o o 3H 3hv •g (H-H )

M (H,Hi) also depends on the orientation of Hi, i.e., the

angle x- Hence the integration over x applies to M (H,Hi) only:

TT

<M^> = Jdx M^(H,Hi) . (II.7)

o

In section II.6.c it is shown that after integrating

analytically, the expression for <M ' contains the same matrix

elements as those required for calculating M (H,Hi) for a

single direction of Hi. Thus,

^ ^ ,3H ,

Q(H) = d<J)

o

de sine-y<M >'UTr^ -g (H-H ) . (ii.o)

a

Eq.(II.8) IS the basis of all our calculations on powder

spectra.

II.2 QUALITATIVE INTERPRETATION OF A POWDER SPECTRUM

A Simple way to simulate a powder spectrum is to calculate

smgle-crystal spectra for a large number of orientations and

to superimpose them m order to obtain Q(H). However, m order

to obtain a reasonable signal-to-noise ratio, such a

calculation requires a very large number of orientations, and

consequently an excessive long computation time.

Eq.(II.8) becomes more transparent when it is rewritten as

Q(H) = y d4) de

sine'<M^>-3H

3hv

•g (H-H ) (II.8a)

a a

This means that instead of integrating the full spectrum over

different orientations of the magnetic field, the integration

IS performed for each line separately. Ofcourse, the

integrand of Eq.(II.8a) is a continuous function of 6 and tj),

although it may be defined m only a part of the e, (J) space.

In Fig.II.3 and II.4 the resonance positions of the various

lines are shown as a function of e , illustrating the more

precise meaning of the index a used m Eq.(II.8a).

By applying some approximations to Eq.(II.8a) one can get a

qualitative idea of the powder spectrum and its relationship

^ 5 = 3^2 g=2 00 D=037crrr^ Vr900GHz r=:60G 3 U 60 30

-1

/'Y

ii 11 1 ^^^^ _...,——-"^ ^ --^ ^ /

—-f-^

\

, ^WT ,

in

0 1 2 3 A 5 6 7 MAGNETIC FIELD (K gauss)

F i g . I I . 3 Calculated powder spectrum of a system with axial

symmetry (a), and the angular dependence of the

resonance positions (c). The schematically integrated

spectrum (b) shows the connection between (a) and (c).

w i t h t h e s i n g l e - c r y s t a l s p e c t r a . F i r s t i t i s n o t i c e d t h a t t h e

s h a p e of t h e powder s p e c t r u m i s h a r d l y a f f e c t e d by t h e a n g u l a r

dependence of t h e l i n e w i d t h ; h e n c e , i t i s r e a s o n a b l e t o

assume T t o be a c o n s t a n t . Second, i n g e n e r a l , <M >|3H / 3 h v |

d o e s n o t show a s t r o n g a n g u l a r d e p e n d e n c e . A l t h o u g h t h i s may

l e a d t o some d e v i a t i o n s i n d e t a i l s , t h i s a n g u l a r d e p e n d e n c e

1 1 / X X I

0 1 2 3 4 5 6 7 8 9 10 11 12

• > mognetic field (kG)

F i g . I I . 4 C a l c u l a t e d powder spectrum of an S=3/2 system with orthorhombic symmetry ( a ) , and t h e angular dependence of the resonance p o s i t i o n s (b) ; f u l l l i n e s for i>=0 d o t t e d l i n e s for (j)=90 . P a r a m e t e r s : g=2.00, D=0.37 cm" , E=0.037 cm" , V=9.00 GHz, r=100 G.

newly d e f i n e d f u n c t i o n S ,

77 71

S^(H) = Jd^ de s i n e g (H-H ) ,

o o

Eq. (II.8a) simplifies to

I 3H

Q(H) = y <M >• h r ^

a a

S (H)

a

II.9

11.10

For the powder spectra under consideration g^(H-H^) is

close to a 6 function, differing from zero only m a narrow

range H -r„ < H < H +r„ . The quantity S (H) now be

interpreted as the area of a narrow contour at the surface of

the unit sphere corresponding with those e and <i> values for which H

a

i) - H (see Fig.II.5). Thus S (H) is a measure for

the number of orientations (f )) for which H H. The derivetive spectrum is obtained as

dQ dH (H) y <M > • a )H 3hv dS 0 'dH -(H) ( I I . 1 0 a ) T h i s m e a n s t h a t t h e p e a k s in t h e d e r i v e t i v e s p e c t r u m a r e f o u n d a t the t u r n i n g p o i n t s m t h e a n g u l a r d e p e n d e n c e o f t h e r e s o n a n c e p o s i t i o n s H , w h e r e S (H) c h a n g e s f r o m a f i n i t e v a l u e to z e r o . T h i s is i l l u s t r a t e d in F i g . I I . 3 , w h i c h s h o w s t h e a n g u l a r d e p e n d e n c e o f t h e r e s o n a n c e p o s i t i o n s a n d t h e r e s u l t i n g f i e l d - s t r e n g t h d e p e n d e n c e o f t h e a b s o r p t i o n o f t h e i n d i v i d u a l t r a n s i t i o n s (the l a t t e r o n l y s c h e m a t i c a l l y ) o f a s y s t e m w i t h a x i a l s y m m e t r y . A l s o s h o w n is t h e p o w d e r s p e c t r u m o b t a i n e d v i a E q . ( I I . 8 a ) w i t h o u t a n y a p p r o x i m a t i o n b u t w i t h a f i x e d I m e w i d t h r = r . A s e x p e c t e d , t h e p o s i t i o n s o f t h e p e a k s in t h e s p e c t r u m c o r r e s p o n d p e r f e c t l y w i t h t h e t u r n i n g p o i n t s m t h e a n g u l a r d e p e n d e n c e o f t h e r e s o n a n c e p o s i t i o n s . T h i s c o n c e p t w a s u s e d in 1964 by V a n R e i j e n (3) m a n a l y z i n g o r t h o r h o m b i c C r ( I I I ) s p e c t r a . T o e s t a b l i s h w h i c h t u r n i n g p o i n t in t h e a n g u l a r d e p e n d e n c e o f t h e r e s o n a n c e p o s i t i o n s c o r r e s p o n d s w i t h w h i c h p e a k i n t h e e x p e r i m e n t a l p o w d e r s p e c t r u m , o n e w i l l o f t e n n e e d a n e s t i m a t e of t h e p e a k h e i g h t s a n d w i d t h s o f t h e l i n e s . T h e p e a k h e i g h t in t h e d e r i v a t i v e s p e c t r u m c a n b e o b t a i n e d b y t h e a p p r o x i m a t i o n t h a t d S (H )/dH is p r o p o r t i o n a l t o S (H )/r w h e n (H o' o' 'o p e a k h e i g h t a • o ) i n d i c a t e s t h e t u r n i n g p o i n t . H e n c e : 3H H , a , <M >• a a 3hv

•s (H )/r„ .

a o H,a II. 11)

As discussed m section II.6.4 s (H ) can be expressed in

a o '^

terms of the second derivatives of the resonance position

with respect to the orientation angles. For axial symmetry,

,2l

'7

'•h

sol ^^fisr^ 4 0 / s o / , '^•STiS'Si a s ^ ?^-r5fe

/ V $ i

/ ^yC

TS

w /

/ W i

/ / T

T

"

t

^ 0 V S K V N > \

j |

I\W

1

i«

T "

_ _ 1

-r — — K" J iS<'iS^ i \ i O

\ ^ °

lOyai »'3^^Er \ B 0

\

_ _ ^ 7 0

\

aJ-^^T 0 10 20 30 40 50 60 70 80 90 100 110 120 130 '40 150 160 170 bO/

/fy

/ " ~ /

-8oL__/_^ 9ol (.0/

/

/ ^

3 0 /

^ / <

< / "

1

iJ'

^^^m.

1

y/yjJjT

/rJ/nJ

rhMA

7r-/-~U

L i T j ^

-JJTJZ

__r~f~

nr"'"

h

1

x>C>^

^

T ^ < \ V x

rf^vVC

^'

J

' =

mv

M

_U

1

i = . . . =

Y

^

-N?

Xx^

V

\ \

\ 4 0

x x°

, / \ XfiO

\^ \ ^A'"

\ \ _ _ _ _ i B O 1 1 m a 10 20 30 ^0 50 60 70 80 90 100 110 120 130 140 150 160 170 130 f » 5 0 / 7o/ / 901 4 ^ 3 0 /

,

i

T ^ ^ t

7-J.

p U

. . .

1

__10

m

1

- - U

^ ^

^^\ 3\™

'\ \ ..A'"

\ ' \ \ Pfs 1 1 90 10 20 30 40 50 60 70 90 100 110 120 130 140 150 160 '''0

Fig.II.5 Polar diagram of the and values at which resonance takes place in the examples of Fig.II.3 and II.4. Three cross sections for constant magnetic field:

(a)H=5.8 kG, (b)H=6.2 kG, and (c)H=6.57 kG; r=100 G; dashed and cross-hatched curves correspond to Fig.II.3; solid and vertical-bar curves correspond to Fig.II.4.

and for less than axial symmetry, ,2H ^ 2 H ^S^H ; ( H ) a o sine, 3^H 3"H a a 36' 34)' )e3(|) (II.lib) a s s u m i n g t h a t t h e c o n t r i b u t i o n s t o S (H ) come from a ^ a o r e l a t i v e l y s m a l l a n g u l a r r e g i o n , and t h a t 6 7^ 0. For t u r n i n g p o i n t s a t e =0 t h e e x p r e s s i o n s f o r S a r e much more c o m p l i c a t e d .

Table I I . 1 Comparison of estimated and c a l c u l a t e d i n t e n s i t i e s of t h e peaks in t h e epr spectrum of F i g . I I . 3 .

I3H /3hv|<M > ' a ' a |3H /3hv|<M >sine |3^H /se^ ' a ' a o' a mtemsity 3H /3hv (kG/GHz) « -1 3H / 3 D (kG/cm ) 3H /3E (kG/cm~S 134 97 97 0.20 -0.30 +8 151 25 29 0. + 10 -16 27 340 30 32 0.41 +9 -38

* Peak height from the calculated powder spectrum of Fig.II.3. For the example of Fig.II.3 some chracteristic quantities of the turning points are given m Table II.1. It is seen that the peak intensities predicted by Eq.(II.lla) agree very well with the intensities obtained by a full calculation via

Eq.(II.8a). From Eq.(II.6) and the quantities of table II.1, the following relations can be derived for the I m e w i d t h s of the example given:

+ 8 r„ r, = 0.20 r + A o 0.3 r, r„ = 0.27 r + 10, B o r^ = 0.41 r + 9. c o + 16 r, r^ + 38 r, Thus undoubtedly: ^A ^ ^B <

'C-When the crystals are perfect i.e., r_ ~ 0 and r „ ~ 0, then r, < r„ « r„. when the crystals are not perfect_{A B C} V « V^

Generally,the analysis of a powder spectrum will be concentrated on the determination of the parameters of the spin Hamiltonian. This IS rather easily done by fitting the calculated magnetic field strength at the turning point m the angular dependence of the resonance position to the experimental line position, provided it is known which turning point corresponds with which line. Comparison of the experimental line intensities and

I m e w i d t h s with the theoretical guesses according to Eqs.TII.ll) and (II.6) will be very helpful m assigning the experimental lines. But when the symmetry is less than axial and the number of lines IS great, the assignment is often hardly possible. In such cases a full simulation of the spectrum will be required to make a reliable assignment.

II.3 COMPUTATION OF THE FULL POWDER SPECTRUM

The powder spectrum is formally described by Eq. II.8 :

i a

^

u

.„

Q(H) = d<p de sine- y <M >• ' a o o

3hv

•g (H-H )

For any direction of the magnetic field, i.e., 6 and <}>, the calculation of H , <M >, 3H /3hv and r„ is straightforward.

a a a H,a Some details of the calculation of these quantities are discussed m section II.6. In this section we are concerned only with the integration.

First It should be noted that the given integration limits are for the most general cases; often the symmetry of the spin Hamiltonian allows a reduction of the integration limits, as indicated m Table II.2.

The simplest way to calculate Q(H) is summing the integrand for a closely spaced set of 6 and i}' values, i.e.,

N M

I I

1=1 3=1 a

Q(H) = I I ysine^-<M^>.

3H I • ^ -g (H-H ) 3hv ^a a

with 6 = ( T T / N ) I and <i) = ( T T / M ) 3 . However, to get a reasonable

spectrum one will need thousands of orientations. Calculation of the spectrum for such a large number becomes prohibitive

Table II.2 Integration limits for the various

centrosymmetric point groups {*)

^2 ' S '

S h ' S ' S h '

°2h'°2'<=2v'

^4h<^4'^4'

V ' ° 4 ' ^ 4 v ' ° 2 d '

^6 ' S '

D . , , ( D , c ) 3 d 3 3 v C^, ( C ^ , C , , ) 6 h 6 3h 6 h 3h 6 v 6 h 6 T, (T) h O, ( T , , 0 ) h d D . (C ) con ocv 0-TT 0-7T/2 0-7T/2 O-Tr/2 0-7V/2 O-TT O-TT/2 0-71 0-7T/2 0 - 7 7 / 2 0 - 7 7 / 2 0-77 0-77 0 - 7 7 / 2 0 - 7 7 / 2 0 - 7 7 / 4 0 - 7 7 / 3 0 - 2 7 7 / 3 0 - 7 7 / 6 0 - 7 7 / 3 0 - 7 7 / 3 0-7T/6 a s S^ o r D ^ ^ ^ ^ ° 3 d ° ^ ° 4 h 0 - 7 7 / 2

* The corresponding noncentrosymmetric point

groups are given within parentheses. e=0

corresponds with the principle axis, e-77/2,

(t)=0 corresponds with a twofold axis perpendicular

to the unique axis (when there i s no such twofold

axis, I t IS irrelevant where to s t a r t the

integration over (}>) .

when t h e syiTimetry i s l e s s t h a n a x i a l and t h e I m e w i d t h i s

s m a l l r e l a t i v e t o t h e m a g n e t i c f i e l d r a n g e c o v e r e d by t h e

s p e c t r u m . To o b t a i n a s i g n a l - t o - n o i s e r a t i o of a b o u t 100, f o r

an a x i a l s p e c t r u m t h e number of o r i e n t a t i o n s r e q u i r e d i s a b o u t

t e n t i m e s t h e r a t i o of t h e f i e l d r a n g e t o t h e l i n e w i d t h . For

a s t r o n g l y o r t h o r h o m b i c s p e c t r u m t h e r e q u i r e d number of

o r i e n t a t i o n s i s a b o u t 500 t i m e s t h i s r a t i o .

A way o u t of t h i s problem i s o b t a i n e d by c o n s i d e r i n g t h a t t h e

a n g u l a r d e p e n d e n c e of t h e r e s o n a n c e p o s i t i o n H and t h e v a l u e

of <M > I 3H /3hv I IS r a t h e r smooth. T h i s means t h a t when H and

_{a ' a ' a}

<M >|3H / 3 h v | a r e known f o r a l i m i t e d number of o r i e n t a t i o n s ,

extending over the complete integration range. It appeared to

be difficult to obtain interpolations going smoothly through

the known points without spurious oscillations. In this section

a short discussion of the results of this search is given. In

the next section a "local interpolation scheme" that is suited

better for calculating powder spectra is discussed separately.

First our experience has shown that the interpolation function

must fulfill two important requirements: serious discontinuities

are only allowed m third and higher derivatives of H vs 6 and

(J), and the number of extrema must be equal to the number of

extrema of the function that interpolates linearly

between neighbouring points (no spurious oscillations). A

consequence is that all types of polynomial are unsuited

because the higher-order terms tend to oscillate, while without

the higher-order terms they are not flexible enough.

Much attention has been paid to the "splme functions", which

have the form

F(x) = I c^-s^(x),

1

in which the "spline" s (x) is (x-a ) for x > a and zero

•^ 1 1 1

elsewhere. The a 's are a set of "knots", more or less

1

uniformly distributed over the range of x under consideration.

The constants c are available to fit the interpolation

function F(x) to the known points. The spline functions are

very flexible, but still show a strong tendency to oscillate,

although much less than polynomials. Many modifications of the

splme function are possible that may remove this problem. A

very simple solution is provided by the "rounded ramp" function

introduced by Lagerlov (4), which have the form

f^(x) = ln[l + exp{b^(x-a^)}),

with a again a knot. This function shows a smooth transition

1 ^

from f (x)=0 for x '^ a to j (x)=b x for x >^ a ; the

•' 1 1 •' 1 1 1

smoothness of this transition may be adapted by the choice of

the b 's. Interpolating with these rounded ramp -Functions

gives perfect axial spectra starting from H and <M >|3H /3hv|

values calculated for about 30 orientations.

All these interpolation methods, however, still have one

common and important disadvantage: it must be established which line at one orientation of the magnetic field corresponds with which line at an other orientation, m the sense as discussed

m the previous section. From Figs. II.3 and II.4 it is seen that the correlation is rather obvious. Yet it is not simple to make a reliable computer program that establishes this

correspondence. Of course one might sort the transitions without resorting to the computer by using a plot of the angular

dependence of the resonance positions as presented m Figs.II.3 and II.4. For S=5/2, however, this is a cumbersome task even if symmetry is axial, and even much more so for lower symmetry.

II 4 INTFRPOLATION SCHEME FOR THE SIMULATION OF POWDER SPECTRA

To be less dependent on the establishment of the correspondence between the transitions, we have chosen a piecemeal

interpolation. This can be realized by requiring that the interpolation function not only describes the resonance positions at the two subsequent orientations involved, but also describes their derivatives, i.e., 3H /3e and 3H /3di.

a a The continuity of the derivatives thus obtained guarantees a smooth interpolation over the whole angular range. For those intervals where the correspondence between the transitions cannot be established completely reliably, the interpolation is replaced by a direct calculation from the spin Hamiltonian of H and <M >I3H /3hvI m a very dense net of orientations. Thus the advantage of the interpolation is lost only m a small angular range.

The quantities H , 3H /3e, 3H /3<}), <M > | 3H /3hv| and the ^ a a a a ' a '

energy-level numbers i and 3, obtained by numbering the energy levels m order of increasing energy, are calculated for all transitions at the corners of a network of orientations. Because corresponding transitions always have the same

M__^^-• H

Fig.II.6 The three elementary patterns in the angular dependence of the resonance positions. Roman numerals refer to examples in Fig.II.3.

same combination 1,3 occurs more than once at the orientation under consideration. For these orientations correlations must be established for sets of neighbouring points of the network. The computer must recognize the different patterns of the angular dependence of H to be met in practice. In Fig.II.6 the three elementary patterns are shown for one variable

angle. The actual patterns can be any combination of them. When only one transition is found at one orientation and none at the other orientation, it is likely that one has to do with pattern III. A check on this is that H + 2(3H /3e) (62-61) is outside the range of the magnetic field considered. When at both orientations only one transition is found, it is likely that one has to do with pattern I. A check on this is

\(S^^ - «a(0'^ = 1(^(9.) + ^(e,)].(6.-6i).

When two transitions are found at one orientation and none at the other one it is likely that one has to do with pattern II. A check on this is (H -H.) ex R dH c 36 3H "36" 3i

In analyzing more complicated patterns one has two additional criteria: the transitions will never cross each other, and the transition probability will not change strongly in a pattern of type I (contrary to a pattern of type II) . Now for each of the patterns of Fig.II.6 interpolation functions are required, while more parameters should be available for H^^,

of which the derivatives must fit too, than for quantities such as <M >, <M >I3H /3hv| and ?„

a a ' a ' H, a

The following fuctions have been chosen: Pattern I: ^ ( e ) =ao+a, e+a2 6 =^+a3 6 ^

" ( II 121 <M >(e)=bo+bie.

Pattern II: _{H (6)=ao+a2e±/aT+a7e,}

Ct ^ P f T T 1 1 '

<M >{e)=bo+bi/a7+a7e-

i±^^-^-^-ex, p

These functions ensure that for

e=-ai/a5=e the transitions collapse:

^ ' max '^

H (6 )=H„(e ) , <M >(e )=<M.>(e ) , a max B max a max 3 max 3H 3H —l i Q )=- P(e )=oo. 36 ^ max^ 36 ^ max^ Pattern III: H (6)=ao+a2( <M >(e)=bo. (11.14:

Extending the scheme to less than axial symmetry, one should make a choice of the system of orientations used for

interpolation. There are two reasonable alternatives: a triangular net of points uniformly distributed over the unit sphere, or a square net of points uniformly distributed over the e, (j) plane. The former is likely the most efficient one, while the latter can be implemented easiest in a computer program. Therefore m this preliminary work we have chosen the square net.

To determine the interpolation function, describing the resonance positions and their derivatives 3H /3e and 3H /3(|) at the four orientations, many combinations of the elementary patterns shown in Fig.II.6 should be taken into account. In our computer program only those situations are considered in which only pattern I occurs with variation of cji, while all three patterns may occur with variation of 6. This means that the computer program is adapted very well to systems with syiTimetry slightly less than axial, but that it will be less efficient

a o = a i = a 2 = 3 3 = bo = b i = a o o + a o 1 tl^+ao 2 ^ + a o 3 a i o + a i i t | ) + a i 2 t | ' ^ + a i 3 3 2 0 + 3 2 1 <i>, 3 3 0 + 3 3 l<i>, b o 0 + b o i<i>, b 1 0 + b 1 1 (j).

make the program somewhat lengthy.

The interpolation functions required for the patterns considered in our computer program can be obtained from Eqs.(11.12), (11.13) and (11.14) by the substitution

, 3

:ii.i5;

T h e s e f u n c t i o n s w i l l g u s r a n t e e t h a t H is c o n t i n u o u s o v e r the a

w h o l e a n g u l a r r a n g e . B u t also w h e n 3H /3e is fitted exactly a t t h e o r i e n t a t i o n s b e t w e e n w h i c h is i n t e r p o a l t e d , it w i l l b e 3 c o n t i n u o u s f u n c t i o n o f (|) b u t n o t n e c e s s a r y of 6. T h e r e v e r s e a p p l i e s t o 3H /3(t). T h i s is a s e r i o u s p r o b l e m o f t h e s c h e m e , w h i c h w i l l i n t r o d u c e some " n o i s e " in the c a l c u l a t e d s p e c t r u m . H o w e v e r , t h i s e f f e c t is r e d u c e d b y t h e f a c t t h a t 3H /36 b e c o m e s c o n t i n u o u s w h e n 3H /3d) V 3 n i s h e s 3nd v i c e v e r s a , thus m those a a n g u l a r r e g i o n s m w h i c h t h e r e s u l t i n g s p e c t r u m is m o s t s e n s i t i v e to d i s c o n i n u i t i e s .

II.5 APPLICATION TO THE SIMULATION OF AN EXPERIMENTAL SPECTRUM

T h e s u i t a b i l i t y of t h e p r o g r a m for the s i m u l a t i o n of e p r p o w d e r s p e c t r a is i l l u s t r a t e d c l e a r l y in F i g . I I . 7 . T h i s f i g u r e shows the e x p e r i m e n t 3 l a s w e l l a s the c a l c u l a t e d s p e c t r u m o f F e ( I I I ) s u b s t i t u t e d a t the a l u m i n i u m s i t e s in A l O O H ( d i a s p o o r ) . The p o w d e r s p e c t r u m h a s b e e n r e c o r d e d by V a n D i 3 k ( 7 ) . T h e a n g u l a r d e p e n d e n c e of the r e s o n a n c e p o s i t i o n s is v e r y c o m p l i c a t e d , r e s u l t i n g m many w e a k h i g h - f i e l d l i n e s . T h e s p e c t r u m can b e d e s c r i b e d by the spin H a m i l t o n i a n (8) H = g6H'S + DS^ + E(S^-S^) + bo f 35s'*-(30S^ + 30S-25) S^ } z X y ^ z z' + b2{(7S^-14S -S^-S+9)S^ + S ^ ( " " ) } + b^CS^+s"), _{' z z + ' +} -g= 2.000 bo=-0.311xlO~'' cm"'^ D=-0.3657 cm""* b2= 4.693xlo"^ cm"' E=-0.1723 cm""' bi,= 4.167x10"'' cm"'.

6 8 10 12 14 16 18 20

-* MAGNETIC FIELD (K GAUSS)

Fig.II.7 X-band powder spectrum of Fe{III) m AlOOH (diaspore). The broad line at 8 kG in the experimental spectrum is due to the apparatus. For clearity the part above 2.5 kG has been magnified by a factor four. The two spectra

In calculating the spectrum an orientation-independent linewidth of 80 Causs was used. The original grid, for which resonance positions, derivatives and intensities had been calculated from the Hamiltonian, consisted of about 1200 points, obtained by deviding the 6 = 0 to 90 into 30 intervals and <t)=0 to 90 into 40 intervals. Interpolation created a grid with intervals

A6=0.2 and Ac!)=0.2 , containing 2x10 orientations in one octant. In about 10% of the cases no interpolation was applied because of dubious correlation of the resonances, or not availability of the appropri3te interpol3tion formul3. Thus direct

cslculation of resonance positions and intensities at some 20,000 extra orientations was necessary.

The time required for computation of this type of spectrum will depend on the computer system used and details of the algorithm, as well as the magnitude of the effective spin, the I m e w i d t h , and the symmetry. Our computer program for axial symmetric

systems takes about (2S+l)^/r seconds ( m which V is the

line-width m Gauss) on an IBM 370-158. The spectrum of Fig.II.3, for example, took 20 sec. When symmetry is less than axial, computation time increases considerably. For example an axial

spectrum for S=5/2 and r=50 G seldom takes more than 3 minutes, while the spectrum of Fig.II.7 took almost 60 minutes. The

latter figure certainly could be better than halved by increas-ing the efficiency of the program.

II. 6 APPENDIX COMPUTATIONAL DETAILS

In this appendix some details of the computation of some quantities used in this chapter are discussed.

II.6.1 RESONANCE POSITION

For an arbitrary choice of the parameters of the spin Hamiltonian and the direction of the magnetic field one can calculate the matrix ff(H), which depends on the strength of the magnetic field only. By solving the eigenvalue equation

one obtains the energy levels E and the eigenvectors C as 3 function of H. A transition between the ith and the 3th energy level will occur 3t that value of the magnetic field at which

X^ (H) E |E (H) - E (H)I - hv = 0 (11.17) This equation is usually solved by C3lculating X (H) for a

series of values of the magnetic field. When X (H )-X (H ^,) < 0

13 n 13 n+1

there will be a value H of the magnetic field between H and

a n H , for which X (H )=0; this value can be approached by

successive interpol3tion. This method h3s been described by, among others, Tynan and Yen (5). A disadvant3ge of this method IS thst if there is more than one value of H between H 3nd

a n H ,. for which X (H )=0, one will overlook some of these

n+1 13 a

solutions. Some of the omitted resonance positions will be retrieved by the interpolation scheme proposed in section II.4. A direct method of calculating all resonance positions as the

(real) eigenvalues of a generalized eigenvalue equation has been given by Belford et al. (6). Thus far, however, no efficient numerical method is available for solving this generalized eigenvalue equation.

II.6.2 DERIVATIVES OF THE RESONANCE POSITION

To obt3in expressions for the derivatives of the resonance

positions with respect to the parameters m the spin Hamiltonisn (including the angles describing the orientation of the magnetic field) we need the expressions for the derivatives of the energy levels, which are discussed first.

The derivative of the energy levels can be obtained via a perturbative expansion of the eigenvalue egu3tion (11.16).

To this end H, C 3nd E are expanded into a Taylor series

with respect to the parameters p. For example, 3E_ 3^E

Ap + ••• + i (Ap)^ + (11.18) 3p2

= E° + ^ Ap + ••• + i "^-^2 1 8p 2 ^ 2

This means that the coefficient of each term in the series must

vanish, i.e.,

(iy'-E") c° = 0

9C ran 9E ,

1 , I8fl ll ^0 _

(fl°-E°) -rr^ + \~- - - ^ C" = 0

1 3p l_3p 3p J 1

Making use of the expansion

3C

-,-^ = y

a C°

3P 3 ^ ^

and the orthogonality property

C°^ff''C° = E» 6

1 3 1 13

one obt

3E^

3p

and

1 ains

= r°^

1

3ff

dp

= c°^ -^

r°

1

^ff

(11.19)

i = C°^ # ^ C° t 2 y (ff. [c°+ |1 c°C°+ M C°l (E -E ) " H

3p3q 1 3p3q ^^^|_ [ i 3p 3 3 3q ij i 3 J

3=E^ (11-20)

The same expression holds for .

3P^

The derivatives of the resonance position can be expressed in

terms of the derivatives of E by expanding A-|E -E | in a

Taylor series:

A = A'' + | ^ A p + | ^ A H + . . - + 4 - ^ ApAH + • • •

3p

"

^

^ 3H 2 3p3H

Implicit differentiation with respect to the parameter p, with

the condition A=A° =hv, gives for the first derivatives

3H |A

^ = - 4 ^ (11.21]

3p 3_A

3H

In the same way, one obtains for the second derivatives with

respect to the angles 6 and cfi under the conditions 3H /3e = 0 and

3 2 H a 363(^1 S^A 3e3tt) 3A 3H

11.22)

and the same expression holds for 3 H /3e and 3 H /Sd) .

Another method of obtaining derivatives of the resonance

positions has been presented in an implicit form by Belford

et al. (10), starting from their "eigenfield" equation.

II.6.3 AVERAGE TRANSITION PROBABILITY

The transition probability for a particular direction of the

static magnetic field depends on the orientation of the magnetic

vector of the radiation, which can have any direction

perpendic-ular to the static magnetic field. For a powder spectrum one

needs only the average of these transition probabilities as

defined m Eqs.(II.2aj and (II.7):

77

<M^> = | d x | < i | / / i ( H i ) | 3 > | ' .

0

U s u a l l y , A/i (Hi ) =BHi'g'S. E x p r e s s i n g t h e d i r e c t i o n s of t h e s t a t i c

m a g n e t i c f i e l d H and t h o s e of Hi m t e r m s of t h e E u l e r a n g l e s

(6, (}), and x) r e l a t i v e t o t h e p r i n c i p l e a x e s of t h e g - t e n s o r ,

H = H ( s i n e c o s i | i ; smesincj); cos6)

Hi = Hi (cosecos(t)Cosx-sin(j)sinx; cosesmcticosx+cosiiisinx; - s m 6 c o s x )

one o b t a i n s

77 <M > = B^Hj I < i I ( c o s 6 c o s ( t ) C O S x - s i n c | ) S i n x ) g S o + ( c o s 6 s i n ( ) ) C o s x + c o s ( | ) s i n x ) g S - s m e c o s x g S | 3 > | ^ dx ( 1 1 . 2 3 ) M a k i n g u s e o f TT 77 W

/cos^X dx = / s m ^ x dx = y and / c o s x s m x dx=0

0 0 o

<M > = (77/2)6^H?{ki Ig cosOcosctS +g cosesin(i)S - g s m e s h > | a 1 I-' ' X ^ X ^y y z z ' '

+ | < i l g sind)S - g coscbS h > | ^ |

' ' X X y ^ Y (11.24)

Clearly the computation of <M> poses no extra problems as

compared with the computation of the transition probability

for a single orientation of Hi, represented by the integrand

of Eq.(11.23).

II.6.4 APPROXIMATION OF S (H ) a 0

An important factor in determining the relative peak intensities

m a powder spectrum is the value of

S (H ) = /d<{i /de sine g(H -H )

_{a 0 J ^ J ^ 0 a}

at the turning point m the angular dependence of the resonance position (H ,6 , d) ]

'^ 0 0 ^ 0

An approximation to S (H ) can be obtained m the following way:

1) The lineshape function is approximated by a normalized square pulse

g(H^-HJ=l/r

= 0

when IH -H < r/2

' 0 a'

otherwise,

2) The angular dependence of H is approximated by

,2u 32jj H = H + ^ a 0 2 _« (( ^H 36'

'o'^ ^ I

a 3(})' (q

f

+

363(1

(6-6^) (*-<}> J ,

3) sine IS approximated linearly.

The expression for S (H ) then becomes

'^ a 0 ^ a ( « o ' - f

d<\> de { s i n 6 + ( 6 - e ) c o s e }

_{^ 1 ' 0 o o'} m w h i c h the i n t e g r a t i o n b o u n d e r i e s a r e g i v e n by (see F i g . I I . 5 ) 3-6 ) 0

<|,(6) = (t)^

-3=^H ai2 'H 3^H

a a

,3^H (6-6 ) ' + 0 3(|)^

•S (H ) ~ sine <

_{a 0 0}

1 3^H fd^H

a a _ I a

3<t)'

-363(1)

111.251

Obviously, this approximation is only correct when a small

angular region contributes to S (H )

^ ^ a o and when

,^0.

In a similar way an approximation to S (H ) can be obtained

when H is independent of <^ (axial symmetry) :

S (H )

a 0

3^H

36'

CHAPTER I I I

S P I N HAMILTONIAN FOR P A I R S OF INTERACTING PARAMAGNETIC IONS

I I I . l INTRODUCTION

The interaction between paramagnetic ions is usually described

by means of a spin Hamiltonian with which the energy levels in

a magnetic field can be calculated. The spin Hamiltonian is a

superposition of single-ion terms and interaction terms. The

former contain spin operators of one ion only, while the latter

contain products of spin operators of both ions, of which the

Heisenberg interaction Si-J-S2 is met most frequently.

EPR spectroscopy is especially applicable to the study of pairs

of orbital singlet ions with a strong interaction. The effect

of spm-orbit interaction then is small. Consequently, the epr

lines will be narrow even at relatively high temperature. For

small spin-orbit interaction effects, as compared with the

the isotropic magnetic interaction, the total spin of the pair

IS a good quantum number and the energy levels split into well

separated multiplets. The spectrum of each of those multiplets

is analyzed most easily with a spin Hamiltonian containing spin

operators acting on the total spin of the pair and describing

only the multiplet under consideration. Contrarily, the spin

Hamiltonian with spin operators acting on the individual spins

describes all multiplets at the same time and consequently

contains more parameters than needed for the description of a

single multiplet.

To compare the magnetic interactions in different systems one

will need the parameters of the interaction terms explicitly

and thus the relations between the parameters of both types of

spin Hamiltonians. Some of the relations were already given by

Owen (1) in 1961. More recently, Chao (2) - by using the

Wigner-Eckart theorem - showed that the relations of Owen are

much more generally applicable than originally stated. Chao's

approach, however, cannot be applied directly to spin

Hamiltonians containing fourth order zero field splitting terms or anisotropic biquadratic interactions. As in this thesis Fe(III) pairs are under consideration, these terms have to be included m the analysis of the epr spectra. Therefore, m this chapter a complete formulation will be worked out.

It will be shown that the application of the properties of irreducible tensorial sets (3) gives rather simple relations between the parameters of both types of spin Hamiltonians m their most general form. These simple relations replace the tedious algebra necessary to calculate the matrix elements of the spin Hamiltonian containing spin operators acting on the individual spins on the basis of the total spin functions of the pair. The latter method was used succesfully among

others by Henning et al. (4), to analyse the epr spectra of

Cr(III) pairs m some oxo-spinels, but its application to the Fe(III) pairs discussed in this thesis is almost unmanagable. Apart from the relations between the two types of spin

Hamiltonians, it is essential first to pay attention to the formulation of the interaction terms. It will be shown that the description of the interaction in terms of spherical tensors is most convenient. The relations mentioned above become simple and the introduction of a linearly independent set of higher order interaction terms (like the anisotropic biquadratic interactions required for Fe(III) pairs) is straightforward.

I I I . 2 FORMULATION OF THE P A I R HAMILTONIAN IN TERMS OF IRREDUCIBLE SPHERICAL TENSOR OPERATORS

The general spin Hamiltonian for two interacting ions in terms of irreducible spherical tensor operators of the individual spins is

H = I

fG"^'(H) +

A^'^'1T^'''(S

) (III.la)

= spin Hamiltonian for ion i, when it has no

interaction with the other ion.

^12= I yB*'"'^^) [T(ki)(Si) X Tf^^'(S2)l (III.lb)

k z ki >• -'

= interaction between the ions.

The most important characteristics of the spherical tensors

are presented in table III.l. For a detailed discussion we

refer to the books of Edmonds (5) and of Fano and Racah (3).

It should be noted that the tensors defined m table III.l

differ by a factor (-2) ^ from those of Fano and Racah; thus

all spin operators are real and unnecessary constant factors

are avoided.

Although some parts of the Hamiltonian of Eq.(III.l) are

usually expressed m terms of cartesian tensors (g-tensor,

J-tensor), application of spherical tensors has the important

advantage that the extensive knowledge of the properties of

the full rotation group (especially transformation under

coordinate rotation, vector-coupling and the Wigner-Eckart

theorem) can be used, which is very important m handling

higher order tensors.

Three examples coming forward on applying the Hamiltonian

of Eq.(III.l), will be used to illustrate these points

a) Any (2S+1)x(2S+1) matrix can be considered to be a

representation of the operator

f t <'T;^^^S)

k=0 q=-k ^ ^

on the basis of the 2S+1 spin functions ls,m>. Then all

(k)

parameters A are linearly independent. This also holds

for the magnetic field independent part of Eq.(III.l) on the

basis of the spin functions | s i ,mi"-|s i ,m2 > .

Table III.l Irreducible spherical tensor operators (k)

T (S) = "contra-standard" tensor operator for spin S T^ ' {S) = T (S) = q-th component of the k-th order

^ ^ "standard" tensor operator, -k£q<_k.

„ T^k) ^ (k) (k) (k) _ (k)* (k)

Rr OY] ~ •'•^^ rotation of coordinates round z-axis by angle y

2nd rotation of coordinates round y-axis by angle B 3rd rotation of coordinates round z-axis by angle a (k)

Dr o \ = standard representation matrix of the k-th irreducible representation of the full rotation group. T5^'''(S)

= (S^)""

S, = S₊ + iS X y

S^,T[J'^(S)

T'"^^ '(S) = (-1)^3 T'_'^^(S) (k) /k(k+l)-q(q±l) T_';';L(S) some i m p o r t a n t t e n s o r o p e r a t o r s q T^'^\s) = (-l)'^ T^^^^ts) ^ q - q 1 1 s^ 1 0 - / 2 S z 2 2 8 ^ 2 1 - ( S S + S , S ) z + + z ' 2 0 / 2 7 3 ( 3 S 2 - S 2 ) 4 4 S | 4 3 - / 2 (S Sl+S^^S ) z + + z 4 2 / r / V { ( 7 S 2 - S ^ - 5 ) S ^ + S ^ ( 7 S ^ - S ^ - 5 ) } 4 1 - / 2 7 7 { ( 7 S ' - 3 S ^ S - S ) S^ + S ^ ( 7 S = ' - 3 S ^ S - S ) } Z Z Z + + z z z 4 0 /2735 • ( 3 5 S ' * - 3 0 S ^ S 2 + 2 5 S 2 - 6 S ^ + 3S'') ' z z z

^'^ ^«B-.^a ^B ••• («'^-- = ^'^'^^

in which a . is a cartesian tensor of rank 2S. The

parameters a n,,t however, are not linearly independent.

Spherical tensors are thus better suited to incorporate a complete set of linearly independent parameters into the spin Hamiltonian.

b) Hi, H7 and Hu m Eq.(III.l) will in general be defined

relative to different coordinate systems {Hi and H2

relative to the syTTimetry axes of ion 1 and 2 respectively;

H12 relative to the symmetry axes of the pair).

Consequently, one has to rotate several tensors to the common coordinate system m which the actual calculation will be performed. However, rotation of a cartesian tensor of rank four - for example - is a rather cumbersome process Although rotation of a spherical tensor is m essence not simpler, one can use the known representations of the full rotation group as indicated m table III.l.

c) The calculation of the matrix elements of spherical tensor operators is much simpler than those of cartesian tensor operators. Compare, for example, <s ,m| s"* | s ,m'> with <s,m|S^Is,m'>.

We shall now discuss the Hamiltonian of Eq.(III.l) in some more detail.

The magnetic field independent part

yA"^'T"^'(S) = y f A'^^T^'^^S)

k k q=-k ^ '^

is the zero field splitting in a conventional form, m which (k)

the A s are the zero field splitting parameters. The field dependent part

y G''^^(H) • T " ^ ' ( S )

k

represents the coupling of the spin S with the magnetic field H. The most general coupling is represented by

y y F^'^''^) flf'^'^x

T'^^(S)

k k ' *•

However, one usually limits the field dependent terms to

k=k'=l. In that case

G(^'(H) T''\S) =F(ll)[Hfl^xTfl'(S)j ,

wich IS equivalent to

H-g-S = yy g^B^a^B (a,6=x,y,z)

aB

G'-'^H) = ^ y H (±g -ig ) + i- a ^ax ^ay

G*^'(H) = -/5 y H g

o i- a ^ a z

_a

The coupling between the spins Si and S2 is described by

Eq.(III.lb). In the next section we have to apply this

operator to the eigenfunctions of the total spin, i.e., to sets

of functions irreducible under rotation of both spins together.

Hence, m order to apply the Wigner-Eckart theorem the direct

products m Eq.(III.lb) must be reduced to operator sets

irreducible under rotation of both spins together.

Hn =111 B^^'-'^'^^^ T^^'-^'^^hsi ,S2) (III.lc)

T;;^='^'^''^^(S/,S2)

-fTtki\si)xTfl^^\s2)l"''

(ki),, ^Jk2),

= y y (ki k2 k q ki qi ;k2 q2 ) T^„ ' '(Si ) T^'"' '(S2 )

3 ' ^ = " ^ ' ^ ^ ' = y ' y ' ( k i k 2 k q | k i q i ; k 2 q 2 ) B ' * ^ ' ^ ^ ' ^ qi q2 ^^ ^^

N o t i c e t h a t b e c a u s e of o u r p h a s e f a c t o r s of t h e t e n s o r o p e r a t o r s ,

t h e a d 3 0 i n t s of t h e c o u p l e d t e n s o r s a r e

^ ( k : k i k 2 ) t ( g ^ ^ g ^ ) ^ (_^)k+ki+k2+q ^(_k^:kik2)(s^^S^)

As the bilinear interaction is usually written as a cartesian

tensor Si'J*S2 we give m table III.2 the relations of this

tensor with the spherical tensors used in Eq.(III.lc) as well

as those used m Eq.(III.lb). Although the operators will not

T a b l e I I I . 2 r e l a t i o n s b e t w e e n t h e c a r t e s i a n t e n s o r n o t a t i o n a n d t h e s p h e r i c a l t e n s o r n o t a t i o n o f t h e b i - l m e a r i n t e r a c t i o n . Si • J - S 2 B 11) T ^ ' ' ( S i ) X T ' ' \ S 2 ) (11) J l l J l l , n J fiS, S , = y y B^^^^ T^^^Si) T^^\S2 f; ^ aB l a 2B '; ^ qi q2 qi q2 B qiq2 (11) ^ ( . 1 ) ^ 1 + ^ 2 B < ' " * q i q 2 - q i - q 2 T f ' ' ( S i ) T f ' \ s 2 ) = ( - 1 ) ^ ' ^ ^ ' T ' ' " ( S I ) Tt'^+(S2) q i q2 - q i - q 2 a,B=x,y,z q i , q 2 = 0 , ± l q i q2 g d i ) _{q i q 2} T ' " ( S I ) T " \ S 2 1 q i q2 1 1 1 0 1 - 1 0 1 0 0 ( 1 / 4 ) ( J - J ) - ( i / 4 ) ( J +J ) XX yy xy yx

= l t S +

- / 1 / 8 ( J - t J ) - / 2 S, S„ xz y z 1+ 2 z - ( 1 / 4 ) ( J + J ) - ( i / 4 ) ( J - J ) -S,_^S„ XX yy xy yx 1 + 2 -- / l / 8 ( J --ij ) --72 S, S^ zx z y I z 2+ ( 1 / 2 ) J , 2S S I z 2z S i . j . S 2 = y B < ' ^ = ' " T ^ ^ = ' ' \ S I , S 2 ) k = l +k ( k : l l ) ( k : l l

l l ^ a , ^ . a ^ 2 , - l I - T ^^

( S l , S 2 ) k=l q = - k ^ ( k : l l ) ^ ^ _ ^ j k t q ^ ( k : l l ) * q - q ( k : l l ) , , , , , k + q ( k : l l ) t , T ( S i , S 2 ) = ( - 1 ) ^ T S i , S 2 ) q - q B (k : 11) T ^ ; - ^ ' \ s i , S 2 ) 2 2 ( 1 / 4 ) (J - J )-{l/4) ( J +J ) XX yy xy yx 2 1 - ( 1 / 4 ) ( J + J ) + ( t / 4 ) ( J +J ) xz zx y z z y 2 0 / l / 6 ( J - ; J - : J ) zz XX yy s s 1 + 2 + - ( S S +S S ) 1+ 2z I z 2+ / i T e (4S S - S , S - S S ) I z 2z 1+ 2 - 1 - 2+ 1 1 ( 1 / 4 ) ( J - J )+{i/4){J - J ) ZX x z yz z y 1 0 i/TJa (J - J ) yx xy ^ z ^ 2 + " ^ + ^ 2 z ' ^ < ^ l - S + - ^ + = 2 - ' 0 0 - / 1 / 1 2 ( J + J + J ) XX yy z z / 1 / 3 (2S S +S S +S S ) I z 2z 1+ 2 - 1 - 2+

From table III.2 the following relations are obvious:

g(0:ll) ^(0:l])jg ^ ^ = J S i - S 2 J=(J +J +J )/3 XX yy zz ' is the isotropic part of the interaction,while

g(l:ll) ^(l:ll)^g^ ^g^j = 0 ( S i A S 2 ) = 1{J -J ) • (S, S. - S , S_ ) xy yx Ix 2y ly 2x + 1{J -J ) • (S, S., - S , S_ ) yz zy ly 2z Iz 2y + i (J -J )• (S, S„ - S , S- ) zx xz Iz 2x Ix 2z is the anti-syrraTietric Dzialoshmsky interaction, and

B ( 2 : 1 1 ) ^ ( 2 : 1 1 ) ^ S i - J ' - S 2 J\=l{J „ + J „ ) - 6 „ J

a g ^ a B Ba aB

IS the anisotropic symmetric interaction.

Inspection of the Hamiltonian of Eqs.(III.l), (III.la) and (III.lb) shows that, apart from the omission of a number of magnetic field dependent terms, it is the most general spin Hamiltonian for the (2Si+1 )*(2S2 + 1) energy levels and that

all parameters are linearly independent. The straightforwardness of the formulation of such a spin Hamiltonian is one of the advantages of the use of irreducible spherical tensors as compared with cartesian tensors. This advantage becomes more apparent when higher order tensors are needed. For example, the full biquadratic cartesian interaction tensor consists of 81 terms of the form S, S, S_ S„ , S, S, S„ S„ . Of these 81 terms

Ix ly 2x 2z Iz ly 2x 2x

only 25 are linearly independent, which are represented by T ' ( S I ) X T •'(S2). As will be discussed m section III.4, of

(2*22)

these 25 terms only the 14 terms m T (Si,S2) and (4:22)

T ' (Si,Sj) will be needed to describe the epr spectra of a pair with a strong isotropic interaction. There is no simple way to extract the required terms from the full cartesian tensor.

Ill 3 APPROXIMATIONS FOR STRONG ISOTROPIC INTERACTION

The isotropic part of the interaction H12 is:

H = y B^°--^^^T''--^^\SI,S2)

° k

:iii.2: When the interaction is strong, i.e., when the differences

m eigenvalues of H are large relative to the effect of

o

H' H-H.

one can treat H' as a perturbation, that only mixes

eigen-functions of H with almost equal eigenvalues. The matrix of

// on the basis of the eigenfuctions of H will become

^ o

"blockdiagonal", which simplifies the calculation considerably relative to the calculation on the basis of the spin product functions |Si mi> |s?m2>. Each block describes the energy levels

(and thus the epr spectrum) of a single multiplet. Thus the epr spectrum is split into a number of less complicated spectra, which can be described independently of each other. Both types of calculations are illustrated in Fig.III.l

31.

X

p

'S.

(S, »S;|^I.. S(S +

F i g . I I I . l Schematic i l l u s t r a t i o n of the effects of the choice of the basis functions on the matrix representation of the spin Hamiltonian of a pair of paramagnetic ions. I t IS s i m p l e t o s e e t h a t t h e t o t a l s p i n f u n c t i o n s :

I Si S2 s m> = y y (Si S2 s m' Si mi ; S2 m2 ) I Si mi > I S2 m2 > mi m2