Mössbauer study of ⁵⁷FE²⁺ ions in some rhombohedral crystals

A thesis submitted for the degree

of Doctor of Philosophy

of the

Australian National University

by

Bat it i y V&nnit> Howza

J

anuary 19 80

The research described in this thesis was

carried out while I was a full-time research scholar

at the Australian National University, and except where

due reference is made is my own.

This thesis contains no material that has been

accepted for the award of any other degree or diploma

in any university or similar institution.

6

.

B.D. HOWES

I w i s h to e x p r e s s my a p p r e c i a t i o n and g r a t i t u d e

to my s u p e r v i s o r , Dr. D.C. Price, for s t i m u l a t i n g d i s c u s s i o n s and h e l p f u l g u i d a n c e d u r i n g this cou r s e of study. I am

also i n d e b t e d to Dr. D. C r e a g h for o f f e r i n g his e q u i p m e n t and time to c a rry out the X - r a y study of the

(M.^ e^ ) ( Py NO ) g ( C104 ) 2 c o m p o u n d s and dis c u s s the r e sults of the analy s i s .

I am g r a t e f u l to Dr. M. W i l t s h i r e and Dr. D. Tay l o r for their i n t e r e s t in my w o r k and also to the t e c h n i c a l

s t aff of the D e p a r t m e n t , p a r t i c u l a r l y Mr. G. S a m p i e t r o for his p r e p a r a t i o n of the Co^ ^ F e ^ C l 2 c r ystals.

F i n a l l y I w o u l d like to thank the A u s t r a l i a n N a t i o n a l U n i v e r s i t y for o f f e r i n g me a Ph.D. s c h o l a r s h i p and also the D e p a r t m e n t of Solid State P h y s i c s for the p r o v i s i o n of l a b o r a t o r y f a c i l i t i e s .

The 57Fe Mössbauer spectra of F e (P y N O )g (C 10 4 ) 2 ,

where PyNO is Pyridine - N - oxide

[viz.

at low temperature in zero applied magnetic field showed

2. "f*

effects of slow relaxation of the Fe ion between its

lowest two (quasi-degenerate) energy levels. The spectra

of small crystals display resolved paramagnetic hyperfine

structure which disappeared when the crystals were ground

to power. These spectra have been reproduced using a

2 +

model in which the Fe sites, which are trigonally distorted

octahedra, experience a small off-axial distortion. The

magnitude of the off-axial crystal field, represented by a

term B 2^ 2 ’ ^s considered to be of the same order as the

hyperfine interaction, so the two were applied together as

perturbations to the coupled electron-nuclear quantum

2 +

system of the high spin Fe ion. Distributions of values

2 2 — 1

of the parameter centred at B n = 0 or 0.03 cm enabled

simulation of the experimental spectra for the unground

2 -1

crystals whereas a distribution centred at B^ = 0.3 cm

was required for the ground crystals. One effect of

grinding the crystals thus appears to be a significant

increase in the average cation site distortion.

2 +

Substitution of Fe ions into the isomorphous

compounds M (P y N O )6 (C 1 0 4 ) 2 (M = Zn,Mg) provided a further

opportunity to study the cation site distortion thought to

d i stortion model mentioned above. The Mössbauer spectra

for the Zn-Fe series indicated that, for a certain

2 -t

c on c e n t r a t i o n range, the proportion of distorted Fe sites is

diminished with respect to F e ( P y N O ) 0(0104)2*

An unusual dependence on the iron concentration was

found for the Mössbauer spectra of the Zn-Fe series. An X-ray

powder diffraction analysis showed that the cell constants

of the Zn-Fe series also exhibit abnormal behaviour as a

function of iron concentration. No such behaviour was

observed for the Mg-Fe series in either the Mössbauer or

X-ray results. The dependences of the Mössbauer spectra and

cell constants on iron concentration in the Zn-Fe series

appear to be correlated and related to changes in the degree

of disorder in the crystal lattice. However, the origin of

this disorder is uncertain.

The solid solution series Co Fe Cl? has also

1 -x x z

been studied. This hexagonal layered system is of particular

interest because of the competing spin anisotropies of the

2+

two cations. The Fe m a g n etisation direction and homogeneity

have been examined as functions of iron concentration at

4.2 K in zero applied field by observation of the hyperfine

interactions at the ferrous site. The results obtained to

date indicate that the hyperfine field direction, and thus

the spin, of the ferrous ion rotates, from its orientation

thought to be a result of competition between the ferrous

2 +

spin anisotropy energy and the Co crystal field

anisotropy energy.

A c k n o w l e d g e m e n t s (iii)

A b s t r a c t (iv)

C h a p t e r 1 G e n e r a l I n t r o d u c t i o n 1

1.1 5 7F e 2+ Ions in the M ( P y N O ) g (CI O4I 2 3 C o m p o u n d s ( M = Fe,Zn,Mg)

1.2 57F e 2+ Ions in C o C l2 7

C h a p t e r 2 T h e o r e t i c a l R e v i e w 12

24

-2.1 E l e c t r o n i c Level S t r u c t u r e of Fe 13 Ions in Sites of T r i g o n a l S y m m e t r y

2.1.1 C r y s t a l Field I n t e r a c t i o n 14 2.1.2 S p i n - O r b i t I n t e r a c t i o n 18

2.2 H y p e r f i n e I n t e r a c t i o n s 19

2.3 M a g n e t i c H y p e r f i n e I n t e r a c t i o n 21 2.3.1 O r i g i n s of the M a g n e t i c 26

H y p e r f i n e I n t e r a c t i o n

2.4 R e l a t i v e I n t e n s i t i e s of A b s o r p t i o n 29 Peaks

2.4.1 S i n g l e C r y s t a l A b s o r b e r s 29 2.4.2 R a n d o m l y P a c k e d P o l y c r y s t a l l i n e 34

Ab s o rb e r s

2.4.3 R e l a t i v e I n t e n s i t i e s of 36 T r a n s i t i o n s b e t w e e n Cou p l e d

E l e c t r o n - N u c l e a r States

2.5 E v a l u a t i o n of C r y s t a l Field P a r a m e t e r s 39 from M ö s s b a u e r S p e c t r a

3.1.1 Crystal Growth 40

3.1.2 Absorber Preparation 42

3.2 Apparatus 4 4

3.2.1 Mössbauer Spectrometer 44

3.2.2 Variable Temperature Controller 46

3.3 Evaluation of Mössbauer Spectra 46

Parameters

Chapter 4 Evidence for Cation Site Distortions 49

in Fe (PyNO)6(ClOy)2

4.1 Introduction 49

4.2 Crystal Sturcture 51

4.3 Results and Discussion 52

4.4 Model for the Distortion of the Cation 56

Site in Fe (P y N O ) s (C1 0 y)2

4.4.1 Derivation of the Coupled 58

Electron-Nuclear Basis States

4.5 Discussion 62

4.6 Conclusions 68

2 +

Chapter 5 Further Studies of the Fe Ion in the 69 Isomorphous M ( P y N O )g (CIO4) 2 Compounds

(M = Zn, Mg)

5.1 Introduction 69

5.2 Results 70

5.2.1 Mössbauer Data 70

5.2.2 X-ray Diffraction Analysis 71

5 . 3 Discussion 74

5.3.1 Mössbauer Data 74

5.3.2 Spin-Spin Coupling between 76

2+

Fe Ions in the M (PyNO) § (C1 01+)2 Compounds

5.3.3 X-ray Analysis 78

5.3.4 Correlation of the Mössbauer 80

and X-ray Results

5.3.5 Effects of Changes in the 82

Crystal Preparation Conditions

5.3.6 Temperature Dependence of 83

the Spectral Features

5.4 Future Work 84

5.5 Conclusions 85

2. ”4"

Chapter 6 Magnetic Behaviour of the Fe Ion in 87

C o C 1 2

6.1 Introduction 87

6.2 Crystal Structure of Co, Fe Cl? 88

6.3 Theoretical Considerations 89

6.4 Results and Discussion 92

6.5 Future Work 97

6.6 Conclusions 98

References 99

C H A P T E R

1

GENERAL I NTRODUCTI ON

The emission and absorption of y-rays without

loss of energy due to recoil of the nucleus is known as

the Mössbauer effect as it was first observed by Rudolf

Mössbauer in 1957 (Mössbauer, 1958). Mössbauer spectroscopy

has since found applications in many diverse fields, such

as solid state physics and chemistry, biology and

archaeology. Its great value derives from the fact that

the width of the emission (or absorption) lines resulting

from transitions from metastable nuclear levels are often

smaller than the interactions between the nucleus and

atomic electrons, the so-called hyperfine interactions.

— i 2

This extremely high degree of resolution (ca. 10 )

has allowed the observation of phenomena which before the

discovery of the Mössbauer effect were considered to be

unmeasurable, for instance, a laboratory measurement of

gravitational red shift and observation of the Zeeman

splitting of nuclear levels. The strength and nature

of the hyperfine interations depend critically on the

electronic, chemical and magnetic state of the atom.

Mössbauer spectroscopy can thus provide considerable

Many nuclides are known to be suitable for use as

Mossbauer nuclides but in the context of this thesis only

57Fe will be considered. Two distinct groups of high spin

ferrous compounds will be examined. Although they have

dissimilar properties they are linked by their common

rhombohedral crystalline structure and consequent

octahedrally coordinated cation sites with trigonal

symme try.

In a cubic octahedral field the free ion ground

multiplet (5D) of the high spin ferrous ion is split into

an orbital doublet and a lower lying orbital triplet

T . Spin-orbit coupling leads to further splitting of

the orbital levels resulting in a triplet ground state.

In the presence of a trigonal field this triplet is split

into a singlet and a doublet which, for the cases of

interest in the present study, is the lower. The

separation of the doublet and singlet is a key factor in

determining the appropriate description for the magnetic

properties of the ion and its characterisation (when

desired) by an effective (or psuedo) spin.

These

of the ion are dependent upon the relative magnitudes of

the trigonal field and spin-orbit interactions.

Two particular cases are pertinent to the present

work. In the first, the trigonal field is much larger

than the spin-orbit coupling, giving rise to a ground

doublet well separated (by ~ 100 cm ) from higher lying

c o u p l i n g are of the same order of m a g n i t u d e w h i c h results in a small s e p a r a t i o n (~ 10 cm ) b e t w e e n the do u b l e t and singl e t . T h e s e cases are c o n s i d e r e d in C h a p t e r 4 and

C h a p t e r 6 r e s p e c t i v e l y . It wi l l be seen that the a p p r o a c h c h o s e n to a n a l y s e the e x p e r i m e n t a l data is g r e a t l y

-i n f l u e n c e d by the d e g r e e of s e p a r a t -i o n of the s p -in^j 1 eve 1 s . F u r t h e r d e t a i l s of the e l e c t r o n i c s t r u c t u r e of a high spin f e r r o u s ion in a t r i g o n a l e n v i r o n m e n t wi l l be g i ven in S e c t i o n 2.1.

1.1 5 7 F e 2 Ions in the M (P y N O ) 6 (C 1 0 4)2 C o m p o u n d s (M = F e , Zn, M g )

2 +

P a r a m a g n e t i c Fe ions in F e ( P y N0)6(C1 0 4)2, where PyNO is Pyridine - N - oxide (viz. C^H^NO), and iron

doped into the d i a m a g n e t i c i s o m o r p h o u s structures

M (P y N O ) e (C 1 04)2 (M = Zn, Mg) c o n s t i t u t e the first group of m a t e r i a l s studied. The i n t e r e s t in these c o m p o u n d s o r i g i n a t e d from an i n v e s t i g a t i o n of high spin ferrous s u b s t a n c e s d e s i g n e d to e x p l o r e the p o s s i b i l i t y that some of them m ight d i s p l a y slow r e l a x a t i o n and, if so, to u n d e r s t a n d the s t r u c t u r a l c o n d i t i o n s r e q u i r e d to allow o b s e r v a t i o n of such p r o c e s s e s .

B e f o r e p r o c e e d i n g f u r t h e r it is of v a l u e to b r i e f l y recall that the e l e c t r o n i c r e l a x a t i o n rates of p a r a m a g n e t i c

2 “f"

spin-lattice and spin-spin interactions tend to disorient

the electronic spin both in time and space. It is a

consequence of these interactions that relaxation rates

are fast in paramagnetic systems. The fast relaxation

causes the magnetic moment of the ion and the resultant

hyperfine field at the nucleus to fluctuate rapidly with the

result that the nuclear spin does not experience a unique

direction of the hyperfine field to precess around.

Consequently, the Mossbauer effect measures a time averaged

field of zero at the nucleus.

Observation of slow relaxation effects in the

• • 2

"I-Mossbauer spectra for Fe ions in ZnC0 3 (Price et ai . ,

1977) and M g C 0 3 (Srivastava, 1976) suggested that other

2 ”{■

substances in which the Fe ions have the same site

symmetry might also show slow relaxation effects. The

cations in these rhombohedral carbonates are coordinated

octahedrally with a trigonal distortion of the octahedron

(point symmetry C .). In the majority of cases for which

* X

ferrous ions occur in distorted octahedral sites the ionic

ground state is an orbital singlet. Spin-orbit mixing

with excited states gives rise to a non-magnetic singlet

ground state and two excited doublets. At low temperatures

only the singlet is occupied and no hyperfine structure

will be visible in the spectra. At higher temperatures

one might expect that Raman relaxation processes within

the ground quintet will allow faster relaxation than is

large excitation energies, A, to states above the ground

doublet. This is the result of a A -2 term in the

expression for the Raman relaxation rate (Price et al.,

1977). It follows that observation of hyperfine structure

in the Mössbauer spectra for such ions is less probable

than for cases with doublet ground states and relatively

large separations from higher lying states (if spin-spin

relaxation is unimportant). Such a system exists in the

carbonates mentioned earlier. A further example is found

in F e ( P y N O ) 6 (C104) 2 •

In the absence of an applied magnetic field the

Mössbauer spectra for F e (P y N O )6(C 1 0 4 ) 2 at low temperature

(Sams and Tsin, 1975a, b; 1976) do not exhibit the

we 11-r e so 1ved paramagnetic hyperfine structure evident

in the spectra of the carbonates. However, application

of small (~ 0.1 T) magnetic fields results in the appearance

of resolved hyperfine structure (Sams and Tsin, 1976). A

similar effect was observed by Price and Srivastava (1976)

24“

for Fe ions in C a C 0 3 and C d C 0 3 . The site symmetry of

the ferrous ions in these carbonates is the same as in

Z n C 0 3 and M g C 0 3 , but to observe resolved hyperfine structure

an applied magnetic field was necessary.

It was suggested by Price and Srivastava that the

doublet ground state was split by £ 0.5 cm and on

this assumption they were able to satisfactorily interpret

their data. Zimmermann et al. (1974a,b) discovered evidence

F e (p a p t )2•C6H6 and tetr a k i s - (1,8- na ph th y r i d i n e ) iron

(II) perchlorate. Both substances have a doublet

- l

ground state which is split by less than 1 cm . The

transition probabilities for spin-lattice relaxation

between the doublet ground states were determined by

Zimmermann to be very small thus enabling the observation

of resolved hyperfine structure in the presence of an

2 +

applied field. In the case of Fe ions in Z n C Ü3

the states of the unsplit ground doublet are magnetic

and resolved hyperfine structure is visible at sufficiently

2 “I"

low temperature whereas for Fe ions in C a C Ü3 the states

of the slightly split ground doublet are non-magnetic and

an external field is required to remagnetise them and allow

the observation of hyperfine structure.

The observations of F e (P y N O )6(C 1 04 ) 2 made by Sams

and Tsin ( 19 75a,b; 19 76 ) and the new results reported

in the following chapters may be well described by a

model in which the doublet ground state is slightly

split by a small off-axial distortion. The splitting is

considered to be of the same order of magnitude as the

magnetic hyperfine interaction and thus even in zero

applied field the split states will be remagnetised to a

small extent by the nuclear magnetic moment. This results

in the onset of hyperfine structure manifested as the

line broadening of the powder spectra of Sams and Tsin

(1975a,b; 1976) and the resolved structure of the many-

The off-axial distortion mentioned above was for

reasons of simplicity assumed to be of rhombic form

resulting from random strain fields within the crystal.

Such a distortion is not expected to be an accurate

r e p resentation but within the limitations of the model

it was anticipated that it would provide a reasonable

insight into the system.

It appears evident from the foregoing that the

2 +

assumption of fast relaxation for high spin Fe ions

is not valid under certain conditions which result in a

doublet ionic ground state well separated from

higher states. In other words it is highly probable that

2 ”4"

the slow relaxation observable in high spin Fe ions

is a consequence of the site symmetry.

1.2 5 7Fe Ions in C o C l2

The randomly mixed two component systems of the

type R M X , where R and M are different magnetic cations 1 “ X X

with competing spin anisotropies and X represents the

anions, have attracted much attention because the differing

characteristics of the constituent cations give rise to

interesting magnetic properties.

Fishman and Aharony (1978) have considered the

concentration versus transition temperature phase diagrams

of alloys of two materials which have competing anisotropies.

They have predicted three kinds of ordered phases and

also a tetracritica 1 point in such systems. The ordered

at either end of the concentration range and to a new

phase, which for a mixture of two antiferromagnets with

different anisotropies has been called oblique-

ant i f er romagnet (OAF)(Matsubara and Inawashiro, 1977), in

the intermediate concentration region. In the OAF phase

there is simultaneous ordering of the two spin components.

The spin of each species of cation has its own axis of

sublattice magnetisation which is directed obliquely to

the easy axis of the pure system. Such a phase is thought

to have been found in, e.g. Co ^ Fe C 1 2 .2H20 (Katsumata

et a l . , 19 79 ) and K2Mn^ x ^ ex ^ 4 (Bevaart et a l . , 19 78).

The solid solution series Co Fe C l 2 is a system l -x x

which has competing spin anisotropies and constitutes

the second group of materials investigated in this thesis.

Anhydrous F e C l 2 and C o C l 2 are hexagonal layered compounds

of the C d C l 2 type in which layers of metal ions are

separated by two layers of halide ions (further details

of the crystal structures will be given in Section 6.2).

Magnetic susceptibility measurements (Starr et a l., 1940)

showed that the susceptibilities of both compounds, from

room temperature to 75 K, obeyed the Curie-Weiss law

(viz. x = C/(T-0) where x is the susceptibility, 0 is the

Curie temperature and C a constant). Extrapolation of these

results gave rise to a positive value of the paramagnetic

Curie temperature which according to the Weiss theory

the occurrence of a ferromagnetic transition. However,

susceptibility measurements at lower temperatures reveal

maxima at approximately the same temperatures as A-type

anomalies observed in the specific heat measurements (Trapeznikowa

and Shubtiikov , 19 35;Trapeznikowa e t a l . , 19 36 ) thus implying

antiferromagnetic transitions. Furthermore, for both compounds a

large fraction of the expected saturation magnetisation can be

produced by magnetic fields that are much smaller than

those usually associated with the exchange coupling in

an antiferromagnet which has a transition temperature

of ~ 24 K (Wilkinson et al., 1959). Materials which

exhibit these unusual magnetic properties are known as

metamagnets (Becquerel and van den Handel, 1939). Below

the ordering temperature of the compound (23.5 K for F e C l2,

24.9 K for C0C I2) the cation spins are aligned ferro-

m a g netically within any one layer while the spins in

alternate layers are aligned antiparallel resulting in an

overall antiferromagnetic structure. The spins are

oriented parallel and perpendicular to the trigonal axis

in F e C l2 and C0C I2 respectively as a result of the dominant

crystal field anisotropy.

There has been some conflict in the literature

concerning the relative magnitudes of the trigonal field

splitting 6 and the spin-orbit coupling parameter A for

2 +

F e C 1 2 • The magnetic properties of the Fe ion are

dependent upon the ratio 6/A and thus it is of importance

to resolve which assessment is more appropriate. Kanamori

that the trigonal field dominated the spin-orbit

interaction (i.e. 6 >> A ) in F e C 1 2 • Thus at low temperatures

a good approximation of the spin system can be given by

an Ising model. This means that the ground doublet of the

2 +

Fe ion is well separated from the higher lying singlet

and consequently the transverse spin components are

completely quenched. Such a model was pursued by other

authors (e.g. Heap, 1962; Yomosa, 1960) to calculate a

number of properties. However, the results of a Mössbauer

study of F e C l2 (Ono et a l. 1964) showed that 6/A ~ 1. In

systems which have 6/A ~ 1 the separation between the doublet

and singlet states is ~ 10 cm and the Ising model, as

described above, is not a valid representation of the magnetic

2 +

properties of the Fe ion. Hazony and Ok (1969) repeated

the Mössbauer analysis and their interpretation of the

measurements was found to support K a n a m o r i ’s contention.

Both sets of authors obtained almost identical temperature

dependences for the quadrupole splitting but their

interpretations of the data were quite different. Ono

et al. derived A = 95 cm 1 , 6 = 119 cm 1 (6/A = 1.25) whereas Hazony and Ok found A = 42 cm , 6 = 340 cm

(6/A = 8.4). Before proceeding further it should be pointed

out that it is possible to obtain such widely differing

deductions from two similar sets of data because the

temperature dependence of the quadrupole splitting does

not offer a reliable or unambiguous method for estimating

static crystal field parameters (see Section 2.5). The

i n d i c a t e s that the c l o s e s t d e s c r i p t i o n of the m a g n e t i c 2 +

p r o p e r t i e s of the Fe ion in F e C l2 is g i ven in terms of a m o d e l in w h i c h the e f f e c t i v e spin is 1 and 6/A ~ 1, in a g r e e m e n t wi t h O n o 's a n a l y s i s of the M o s s b a u e r data.

A m o d e l of this type has be e n used to i n t e r p r e t m e a s u r e m e n t s o b t a i n e d from e.g. n e u t r o n - s c a t t e r i n g , Raman s c a t t e r i n g , the para 1lei and perpendicular

s u s c e p t i b i l i t i e s and the s p e c i f i c heat of F e C 12 by B i r g e n e a u et al. (1972), J o h n s t o n e et a l . (1978),

B e r t r a n d et al. (1974) and L a n u s s e et al. (1972) r e s p e c t i v e l y . All a c h i e v e d good a g r e e m e n t b e t w e e n theory and e x p e r i m e n t .

T a w a r a y a and K a t s u m a t a (1979) have o b s e r v e d , u s ing s u s c e p t i b i l i t y m e a s u r e m e n t s , three d i s t i n g u i s h a b l e

m a g n e t i c a l l y o r d e r e d p h a s e s in C o 1 ^ F e ^ C l2 • They a s s o c i a t e d these phases w i t h the F e - r i c h and C o - r i c h

a n t i f e r r o m a g n e t i c phases, and a p h a s e at the i n t e r m e d i a t e c o n c e n t r a t i o n region (viz. 0.75 £ x £ 0.65 at 5 K) they i d e n t i f i e d w i t h the OAF p h a s e m e n t i o n e d earlier. In the pre s e n t s t udy of Co ^ ^ F e ^ C l2 the b e h a v i o u r of the ferrous spin as a f u n c t i o n of iron c o n c e n t r a t i o n is o b s e r v e d via the h y p e r f i n e i n t e r a c t i o n s at the iron ion. Such an

C H A P T E R 2

T H E O R E T I C A L R E V I E W

In this c h a p t e r o u t l i n e s and d e f i n i t i o n s w i l l be g i v e n only of the e s s e n t i a l t h e o r e t i c a l a s p e c t s w h i c h are r e l e v a n t to the w o r k of this thesis and that p r o v i d e

p e r t i n e n t b a c k g r o u n d i n f o r m a t i o n . In p a r t i c u l a r the d e r i v a t i o n of the e l e c t r o n i c e n e r g y level s t r u c t u r e of

2 +

Fe ions u n der the i n f l u e n c e of a tr i g o n a l c r y s t a l field and s p i n - o r b i t c o u p l i n g w i l l be d i s c u s s e d . F e a t u r e s of the h y p e r f i n e i n t e r a c t i o n s will be c o n s i d e r e d wi t h

p a r t i c u l a r r e f e r e n c e to the d e s c r i p t i o n of the m a g n e t i c h y p e r f i n e i n t e r a c t i o n by the e f f e c t i v e field a p p r o x i m a t i o n . The r e l a t i v e i n t e n s i t i e s of M ö s s b a u e r a b s o r p t i o n peaks

2

T-2.1 E l e c t r o n i c Level S t r u c t u r e of Fe Ions in Sites of T r i g o n a l S y m m e t r y

The M ö s s b a u e r s p e c t r a of ions in a crystal l a t t i c e depe n d v e r y s t r o n g l y on the e l e c t r o n i c energy level s t r u c t u r e of the ions in a p a r t i c u l a r c r y s t a l l i n e e n v i r o n m e n t . In this s e c t i o n it is i n t e n d e d to p r o v i d e an o u t l i n e of the e ffects of the v a r i o u s factors a c t i n g upon the free ion g r o u n d state

2 +

of a Fe ion when it is l o c a t e d in a site of t r i gonal

2 |

-s y m m e t r y . This d i s c u s s i o n w i l l be r e s t r i c t e d to Fe ions in t r i g o n a l sites b e c a u s e of their p a r t i c u l a r r e l e v a n c e to the w o r k p r e s e n t e d in this thesis.

The H a m i l t o n i a n w h i c h d e s c r i b e s the s p l i t t i n g of the free ion ground term ( 5D) may be wr i t t e n :

X = X ^ + W

+ X

+ X + X

ct so ss m q

w h e r e X ^ r e p r e s e n t s the c r y s t a l field i n t e r a c t i o n , Xgo is the i n t r a i o n i c s p i n - o r b i t i n t e r a c t i o n ,

X

ss

s p i n - s p i n i n t e r a c t i o n ,

X

i n t e r a c t i o n and

X^

The i n t r a i o n i c s p i n - s p i n c o u p l i n g will be n e g l e c t e d b e c a u s e in the pre s e n t c o n t e x t it w i l l not lead to f urther s p l i t t i n g of the ionic sta t e s (the only n o n - z e r o m a t r i x el e m e n t s are those d e r i v e d from the spin o p e r a t o r S^) and since it is e x p e c t e d to be small (< 1 cm 1 ; A b r a g a m and Ble a n e y , 19 70).

The last two terms of E q u a t i o n 2.1 o p e r a t e on

the n u c l e a r and e l e c t r o n i c c o m p o n e n t s of the w a v e f u n c t i o n s

i m p o r t a n c e of such terms, p a r t i c u a r l y in the c o n t e x t of this t h e s i s ) w i l l b e c o m e a p p a r e n t in C h a p t e r 4. In this s e c t i o n only p u r e l y e l e c t r o n i c w a v e f u n c t i o n s are c o n s i d e r e d and thus these terms are ignored. The term 3Cs a r i s e s from d i p o l e - d i p o l e c o u p l i n g and e x c h a n g e i n t e r a c t i o n s b e t w e e n the e l e c t r o n spins of n e i g h b o u r i n g ions. Both of these i n t e r a c t i o n s may indu c e t r a n s i t i o n s b e t w e e n spin states,

l e a d i n g to the we l l k n o w n s p i n - s p i n r e l a x a t i o n process. In a p a r a m a g n e t i c c o m p o u n d the e x c h a n g e i n t e r a c t i o n is e x p e c t e d

to be s m a l l and d i p o l a r c o u p l i n g b e t w e e n n e i g h b o u r i n g e l e c t r o n s p ins is not s u f f i c i e n t l y s t r o n g to have any s i g n i f i c a n t

e f f e c t s on the e l e c t r o n i c s t r u c t u r e . T h e r e f o r e , it is

p o s s i b l e to omit these i n t e r a c t i o n s from a d i s c u s s i o n of the e l e c t r o n i c s t r u c t u r e of ions in a p a r a m a g n e t i c compo u n d . W h e n a c o m p o u n d w h i c h is m a g n e t i c a l l y o r d e r e d is c o n s i d e r e d , h o w e v e r , the degeneracy of the e l e c t r o n i c levels may be lifted by the e x c h a n g e i n t e r a c t i o n s . The form and i m p l i c a t i o n s of the p r e s e n c e of stro n g e x c h a n g e i n t e r a c t i o n s will be e x a m i n e d in C h a p t e r 6 for the p a r t i c u l a r case of the a n t i f e r r o m a g n e t i c Co Fe^ Cl^ c o m p o u n d s . In this s e c t i o n only those terms in the H a m i l t o n i a n 2.1 r e p r e s e n t i n g the c r y s t a l field and s p i n - o r b i t i n t e r a c t i o n s w i l l be c o n s i d e r e d .

2.1.1 C r y s t a l Field I n t e r a c t i o n

V ( r , 0 , <J>)

l

2 . 2

w h e r e i s t h e c h a r g e o f t h e j n e i g h b o u r i n g i o n l o c a t e d

a t a d i s t a n c e R f r o m t h e o r i g i n . T h e p o t e n t i a l ma y b e

e x p a n d e d i n s p h e r i c a l h a r m o n i c s u s i n g t h e s p h e r i c a l h a r m o n i c

a d d i t i o n t h e o r e m (e . g . G r i f f i t h , 1 9 6 1 ) a n d w r i t t e n a s i n

E q u a t i o n 2 . 3 ( B l e a n e y a n d S t e v e n s , 1 9 5 3 , H u t c h i n g s , 1 9 6 4 ) :

V ( r , 0 , <J)) I Am < r n > Ym ( 0, ( f ) ) 2 . 3 n , m

w h e r e Ym ( 9 , d ) ) a r e s p h e r i c a l h a r m o n i c s , Am d e n o t e s a c r y s t a l l i n e

n n

f i e l d p a r a m e t e r a n d < r n > i s t h e e x p e c t a t i o n v a l u e o f t h e n ^

p o w e r o f t h e d e l e c t r o n r a d i u s . T h e p r o d u c t s A™ < r n > a r e

n o r m a l l y d e t e r m i n e d b y f i t t i n g t h e c r y s t a l l i n e f i e l d H a m i l t o n i a n

t o e x p e r i m e n t a l d a t a . U s i n g o p e r a t o r e q u i v a l e n t s a n d

f o l l o w i n g O r b a c h ( 1 9 6 1 ) i t i s p o s s i b l e t o r e w r i t e E q u a t i o n

2 . 3 a s :

v , m n _ _ m , _ . ) A < r > 6 0 ( L)

L n n n —

n , m

2 . 4

T Bm 0 m ( L ) u n n — n , m

w h e r e 0 (_L) a r e o p e r a t o r e q u i v a l e n t s , t h e f o r m o f w h i c h i s

g i v e n b y O r b a c h ( 1 9 6 1 ) , a n d B™ = A™ < r n > 0 ^ . T h e m u l t i p l i c a t i v e

f a c t o r s 0 ^ a r e t a b u l a t e d b y H u t c h i n g s ( 1 9 6 4 ) .

p

-j-T h e Fe i o n s c o n s i d e r e d h e r e i n a r e i n c u b i c -

o c t a h e d r a l c r y s t a l f i e l d s w i t h a t r i g o n a l d i s t o r t i o n a l o n g

t h e [ 1 1 1 ] d i r e c t i o n o f t h e o c t a h e d r o n . I f t h e q u a n t i s a t i o n z

w h i c h is a t h r e e - f o l d s y m m e t r y a x i s , t h e n b e c a u s e in the

p r e s e n t c a s e m a n y of the t e r m s in E q u a t i o n 2.3 h a v e zero

m a t r i x e l e m e n t s the g e n e r a l f o r m of V s h o w i n g a d i s t o r t i o n

a l o n g the [ 1 1 1 ] d i r e c t i o n is ( B l e a n e y and S t e v e n s , 1 9 5 3 ) :

V B ° 0 ° + B ° 0 ° + B ^ O * 2 . 5

E q u a t i o n 2.5 is a s u m m a t i o n of c u b i c an d a x i a l

c r y s t a l f i e l d terms.

It is a s s u m e d t h a t the c r y s t a l f i e l d i n t e r a c t i o n

is m u c h w e a k e r t h a n the i n t r a - a t o m i c C o u l o m b i n t e r a c t i o n s ,

so that no a d m i x t u r e of s p e c t r o s c o p i c t e r m s by the c r y s t a l

f i e l d is c o n s i d e r e d . It is a l s o a s s u m e d that the c u b i c

c o m p o n e n t of the c r y s t a l f i e l d i n t e r a c t i o n is m u c h l a r g e r

t h a n a ny of the o t h e r p e r t u r b i n g e f f e c t s . In f act, for

2 *4"

Fe i o n s in o c t a h e d r a l s y m m e t r y in c r y s t a l s s u c h as t h o s e

of i n t e r e s t h e r e the s p l i t t i n g d ue to the c u b i c f i e l d is

~ 10 cm w h e r e a s th a t for the t r i g o n a l f i e l d is < 1 0 J

- l

cm . On thi s b a s i s it is p o s s i b l e to m a k e the

a p p r o x i m a t i o n that the g r o u n d s t a t e in the c u b i c f i e l d can

be u s e d to c a l c u l a t e the s t a t e s r e s u l t i n g f r o m o t h e r

s m a l l e r p e r t u r b a t i o n s . In o t h e r w o r d s , a d m i x t u r e s w i t h

the e x c i t e d c u b i c s t a t e m a y be i g n o r e d in s o m e

c i r c u m s t a n c e s . In this c a s e the g r o u n d 5T s t a t e is

2g

c o n s i d e r e d to be e q u i v a l e n t to a 5P s t a t e w i t h the

r e p l a c e m e n t of by an e f f e c t i v e o r b i t a l a n g u l a r m o m e n t u m

a jL w h e r e £ = 1 and a = -1 ( G r i f f i t h , 1 961; A b r a g a m and

P r y c e , 1 9 5 1 ) . C a l c u l a t i o n s p e r f o r m e d by Sa m s a n d Tsin

i n t o t h e g r o u n d s t a t e ( 5T ) , i n d i c a t e d t h a t s i g n i f i c a n t

2 8

e r r o r s w e r e n o t i n t r o d u c e d b y t h i s a p p r o x i m a t i o n .

T h e t e r m f o r n = m = 0 h a s b e e n o m i t t e d i n t h e

e x p r e s s i o n f o r t h e c r y s t a l f i e l d p o t e n t i a l 2 . 5 b e c a u s e

i t i s a n a d d i t i v e c o n s t a n t a n d d o e s n o t g i v e r i s e t o

s p l i t t i n g o f s t a t e s .

5 2 +

T h e D g r o u n d s t a t e o f t h e F e i o n ( c o n f i g u r a t i o n

3 d 5 ) i s s p l i t b y t h e c u b i c - o c t a h e d r a l f i e l d i n t o a g r o u n d

s t a t e o r b i t a l t r i p l e t ( 5 T ) a n d a n o r b i t a l d o u b l e t ( 5 E )

2g g

s e p a r a t e d t y p i c a l l y b y ~ 1 0 4 cm ( A b r a g a m a n d B l e a n e y ,

1 9 7 0 ) ( F i g u r e 2 . 1 ) . T h e a n g u l a r p a r t s o f t h e 5 T

28

w a v e f u n c t i o n s ma y b e w r i t t e n a s ( B l e a n e y a n d S t e v e n s , 1 9 5 3 )

4>i = ~ (2 / 3>2 Y~ 2 - ( i / 3)2 y '2

K

-h

h

<t>-1 = (2/ 3) y

\

T h e s e s t a t e s a r e s p l i t b y t h e t r i g o n a l f i e l d i n t o

a n o r b i t a l d o u b l e t (<{) + ^ ) a n d a n o r b i t a l s i n g l e t ( ( j ) ^ ) . T h e

3 - 1

s e p a r a t i o n b e t w e e n w h i c h i s t y p i c a l l y < 10 cm ( A b r a g a m

a n d B l e a n e y , 1 9 7 0 ) ( F i g u r e 2 . 1 ) . T h e M o s s b a u e r

q u a d r u p o l e i n t e r a c t i o n d a t a o f Sams a n d T s i n ( 1 9 7 5 a ) a n d

2

-f-Ono et al. ( 1 9 6 4 ) h a v e s h o w n t h a t f o r F e i o n s i n a l l t h e

c o m p o u n d s s t u d i e d i n t h e p r e s e n t w o r k t h e o r b i t a l d o u b l e t

2 . 1 . 2 S p i n - O r b i t I n t e r a c t i o n

E a c h of the o r b i t a l s t a t e s h a v e f i v e - f o l d s p i n

d e g e n e r a c y . T h e s e d e g e n e r a c i e s a r e p a r t i a l l y l i f t e d w h e n

a c c o u n t is m a d e of the c o u p l i n g b e t w e e n the s p i n a nd

o r b i t a l a n g u l a r m o m e n t a . F or the f r e e i on this c o u p l i n g

h a s the f o r m

A

A

o— — o

c o v a l e n c y on an ion s i t u a t e d in a c r y s t a l m u s t be

c o n s i d e r e d , h o w e v e r , a nd m a y be a p p r o x i m a t e d by a f r a c t i o n a l

d e c r e a s e in Aq ( I n g a l l s , 1 9 6 4 ) . T h u s the s p i n - o r b i t

c o u p l i n g m a y n o w be w r i t t e n :

A

A

— — o z z + - - +

w h e r e a 2 is a f a c t o r w h i c h a c c o u n t s for the d e c r e a s e in

2 . 7

A

o

d u e to c o v a l e n c y e f f e c t s .

A p p l i c a t i o n of s u c c e s s i v e p e r t u r b a t i o n c a l c u l a t i o n s ,

w h i c h i n t r o d u c e s u c c e s s i v e l y s m a l l e r t e r m s of the H a m i l t o n i a n

2.1 to the f r e e ion g r o u n d s t a t e , r e s u l t in the e l e c t r o n i c

l e v e l s t r u c t u r e s h o w n in F i g u r e 2.1. It is n o t e d that the

s p a c i n g s s h o w n a r e i l l u s t r a t i v e o n l y a n d do no t a c c u r a t e l y

2m\~ #

r e p r e s e n t e i t h e r of the Fe s y s t e m s s t u d i e d in th i s t h e s i s .

It s h o u l d be s t r e s s e d t h a t th e a b o v e d i s c u s s i o n

of the e l e c t r o n i c l e v e l s t r u c t u r e h a s a s s u m e d th a t the

t r i g o n a l f i e l d i n t e r a c t i o n is m u c h l a r g e r t h a n the s p i n -

o r b i t i n t e r a c t i o n w h i c h c a n th u s be a p p l i e d as a

p e r t u r b a t i o n to the t r i g o n a l s t a t e s . T h i s is, h o w e v e r ,

v e r y o f t e n n o t the c ase. In s u c h i n s t a n c e s the s a m e s t a t e

d e g e n e r a c i e s w i l l r e s u l t as a re i n d i c a t e d for the s p i n -

XL.S

Fig. 2.1 Schematic diagram of the electronic energy

levels of the high spin F e2+ ion in a crystal

field of trigonal symmetry as in F e (P y N O )5(C I O4 )2

and F e C l2. (a) The effect of the cubic and

trigonal crystal fields the 5D free ion state.

(b) The effect of the trigonal crystal field component and the spin-orbit coupling on the 5T

orbital triplet. The values of the parameters

used were for illustrative purposes only o

2

2g

B u = - 55.56 cm 1 , B° = - 125 cm 1 and

- 1

s p l i t t i n g s b e t w e e n states will, of course, be m o d i f i e d d e p e n d i n g on the r e l a t i v e m a g n i t u d e s of the t r i gonal field and s p i n - o r b i t i n t e r a c t i o n s . In a n h y d r o u s ferrous c h loride, for inst a n c e , the t r i g o n a l field i n t e r a c t i o n is of the same o rder as the s p i n - o r b i t i n t e r a c t i o n (Ono et a l ., 1964). In this case the t r i g o n a l field and s p i n - o r b i t i n t e r a c t i o n s are a p p l i e d j o i n t l y to the c u b i c states to d e t e r m i n e the e n e r g y level s t r u c t u r e p e r t i n e n t here.

2 . 2 H y p e r f i n e I n t e r a c t i o n s

The h y p e r f i n e i n t e r a c t i o n s w h i c h o r i g i n a t e from the c o u p l i n g b e t w e e n the n u c l e u s and the atomic e l e c t r o n s

c o n t r i b u t e terms to the total H a m i l t o n i a n for the atom w h i c h may be w r i t t e n :

Jf = JC + (E + M, + E, + h i g h e r order terms) 2.8

o o 1 z

r e p r e s e n t s all terms in the H a m i l t o n i a n for the atom except the h y p e r f i n e i n t e r a c t i o n s . is the e l e c t r i c

m o n o p o l e i n t e r a c t i o n , M x is the m a g n e t i c dip o l e i n t e r a c t i o n and E 2 is the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n .

p r o d u c e d by the e l e c t r o n s w h i c h s u r r o u n d the nucleus.

The e l e c t r i c m o n o p o l e term E , w h i c h r e p r e s e n t s the

o

C o u l o m b i n t e r a c t i o n b e t w e e n the e l e c t r o n s and a point n u c l e a r c h a r g e is not a h y p e r f i n e i n t e r a c t i o n . H o w e v e r , the c o r r e c t i o n to this interaction, r e q u i r e d to a c c o u n t for the o v e r l a p of the e l e c t r o n s w i t h a n u c l e u s of finite

dimensions, is a h y p e r f i n e i n t e r a c t i o n and d e t e r m i n e s the s o - c a l l e d iso m e r shift. As a c o n s e q u e n c e of the d i f f e r i n g c h a r g e radii of the g r o u n d and e x c i t e d levels of the n u c l e u s

the C o u l o m b i n t e r a c t i o n w i t h the e l e c t r o n i c charge is d i f f e r e n t for the two states. The y - r a y e n e r g y is thus c h a n g e d r e l a t i v e to its v a lue for a point n u c l e u s by an a m o u n t p r o p o r t i o n a l to the total e l e c t r o n d e n s i t y at the n u c l e u s . If the c h e m i c a l e n v i r o n m e n t s of the M ö s s b a u e r ions

in the s o u r c e and a b s o r b e r differ, the total s e l e c t r o n d e n s i t i e s at the s o u r c e and a b s o r b e r n u c l e i will also be

d i f f e r e n t . The s u b s e q u e n t d i f f e r e n c e in t r a n s i t i o n e n e r g i e s , the i s o m e r shift AE, has been d e r i v e d by, for example,

W e r t h e i m (1964):

AE = f1- Z e 2 [R2- R 2 ][ 14» <0) I 2 - I tjj(o) I 2 ] 2.9

5 e g ' 1 a 1 's

w h e r e R and R are the e f f e c t i v e radii of the e x c i t e d and

e g

g r o u n d n u c l e a r states, | ip (0 ) | ^ and | ip ( 0) | ^ are the total s e l e c t r o n d e n s i t i e s at the a b s o r b e r and sou r c e nuclei.

m o m e n t r e f l e c t s the d e v i a t i o n of the n u c l e u s f r o m s p h e r i c a l

s y m m e t r y and n u c l e i w h o s e s p i n s a re 0 or \ h a v e a zero

q u a d r u p o l e m o m e n t . T h e H a m i l t o n i a n w h i c h d e s c r i b e s this

i n t e r a c t i o n is w r i t t e n in E q u a t i o n 2 . 1 0 in the f o r m

d e t e r m i n e d by A b r a g a m (1961) and C o h e n and R e i f (1957).

“ f r f e j

+ n<i* - i;>]

2 -10

w h e r e eq = 9 2V / 3 z 2 (i.e. the z c o m p o n e n t of the e l e c t r i c

f i e l d g r a d i e n t ) and r\ is the a s y m m e t r y p a r a m e t e r of the

e . f . g . d e f i n e d by

n = ( 3 2V / 3 x 2 - 3 2V/3y2 y 3 2V / 3 z 2

w h e r e the x , y and z a x e s a r e n o r m a l l y c h o s e n so that

3 2 V / 3z 3 2 V/3x 3 2 V / 3y

a n d h e n c e 0 < n < 1.

T h e m a g n e t i c d i p o l e i n t e r a c t i o n , Mi, is the c o u p l i n g

of the n u c l e a r m a g n e t i c d i p o l e m o m e n t w i t h the e l e c t r o n s .

It is p o s s i b l e , f o l l o w i n g A b r a g a m and P r y c e ( 1 9 5 1 ) , to

r e p r e s e n t this i n t e r a c t i o n in a s p i n - H a m i l t o n i a n form:

S • A- I 2.11

w h e r e A is k n o w n as the m a g n e t i c h y p e r f i n e t e n s o r . Th i s

i n t e r a c t i o n w i l l be d i s c u s s e d in s o m e d e t a i l in the

f o l l o w i n g s e c t i o n .

2 . 3 M a g n e t i c H y p e r f i n e I n t e r a c t i o n

be d i s c u s s e d f u r t h e r w i t h the v i e w to u n d e r s t a n d i n g w h e n o ne

is j u s t i f i e d in d e s c r i b i n g it by the e f f e c t i v e f i e l d

a p p r o x i m a t i o n .

T h e e f f e c t i v e f i e l d a p p r o a c h is a s i m p l i f i e d

d e s c r i p t i o n of the i n t e r a c t i o n that is v a l i d u n d e r c e r t a i n

c i r c u m s t a n c e s w h i c h w i l l be c l a r i f i e d b e l o w . O n e c o n s i d e r s

th a t the e l e c t r o n s i n t e r a c t w i t h the n u c l e u s v i a a m a g n e t i c

f i e l d at the n u c l e u s p r o d u c e d by the e l e c t r o n s . As a

c o n s e q u e n c e of t r e a t i n g the i n t e r a c t i o n in this w a y the

b a c k e f f e c t of the n u c l e a r m a g n e t i c m o m e n t on the e l e c t r o n i c

s y s t e m is i g n o r e d . T h i s m e a n s t h a t the e l e c t r o n i c s y s t e m

is a s s u m e d to be u n a f f e c t e d by the i n t e r a c t i o n an d any

a d m i x t u r e of e l e c t r o n i c s t a t e s is n e g l i g i b l e . T he

f u n d a m e n t a l a p p r o x i m a t i o n m a y be o b t a i n e d f r o m the

g e n e r a l i s e d H a m i l t o n i a n ( E q u a t i o n 2.11) a nd is w r i t t e n :

K = <S •A> . I

m — = —

2.12

and H < S • A>

— e f f — =

w h e r e _H ^ is the m a g n e t i c h y p e r f i n e f i e l d ( e f f e c t i v e

f i e l d ) . T h e c o u p l i n g b e t w e e n the n u c l e a r m a g n e t i c m o m e n t ,

p , and the e l e c t r o n s m a y be r e w r i t t e n : — n

" H r _{:e f f}

2. 1 3

g 3 I * H n n— — eff

w h e r e g is the n u c l e a r g - f a c t o r a nd 3 d e n o t e s the n u c l e a r

n n

m a g n e ton.

the f o l l o w i n g m a n n e r : the m a g n e t i c h y p e r f i n e i n t e r a c t i o n

d o e s n o t c o u p l e e i g e n f u n c t i o n s of the e l e c t r o n i c H a m i l t o n i a n .

In o t h e r w o r d s the h y p e r f i n e o p e r a t o r s do n ot c a u s e a d m i x t u r e

of the e l e c t r o n i c s t a t e s b e c a u s e the o n l y n o n - z e r o m a t r i x

e l e m e n t s of the e l e c t r o n i c c o m p o n e n t of the h y p e r f i n e

i n t e r a c t i o n are the d i a g o n a l o n e s a nd t h e s e ar e e q u i v a l e n t

to an e f f e c t i v e f i e l d a c t i n g on the n u c l e u s . In this

a p p r o x i m a t i o n the e l e c t r o n i c and n u c l e a r s y s t e m s a re t r e a t e d

i n d e p e n d e n t l y of e a c h o t h e r and the w a v e f u n c t i o n s for the

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

n u c l e u s of the ion c a n t h e n be w r i t t e n in the p r o d u c t

form:

<J> (e , n)

^i (e) Xj(n)

w h e r e e , n r e p r e s e n t the e l e c t r o n a nd n u c l e a r c o o r d i n a t e s

r e s p e c t i v e l y . X ^ ( n ) c a n be w r i t t e n in t e r m s of the b a s i s

s t a t e s I I , m > .

If the e l e c t r o n i c l e v e l s e p a r a t i o n is l a r g e c o m p a r e d

- 2 - 1

w i t h the m a g n e t i c h y p e r f i n e i n t e r a c t i o n e n e r g y (~ 10 cm )

th i s a p p r o x i m a t e t r e a t m e n t is p e r f e c t l y v a l i d . H o w e v e r ,

if t h e y a r e of the s a m e o r d e r of m a g n i t u d e t h e r e m a y be

m i x i n g of the e l e c t r o n i c s t a t e s by the h y p e r f i n e o p e r a t o r s

a nd the a p p r o x i m a t i o n w i l l t h e n n o t be a p p l i c a b l e . U n d e r

s u c h c i r c u m s t a n c e s the e l e c t r o n i c a nd n u c l e a r s y s t e m s m u s t

be c o n s i d e r e d as o ne c o u p l e d q u a n t u m s y s t e m w i t h b a s i s

s t a t e s d e f i n e d by lL,m ,S ,m ,I,m >.

' L s I

It is i m m e d i a t e l y a p p a r e n t that the s i z e s of the

of the coupled electron-nucleus system are much larger than

in the corresponding cases for which the effective field

approximation may be applied and only the nuclear matrix

elements are required. Nevertheless, for cases in which

the hyperfine coupling between well separated electronic

states may be ignored the computational difficulties can

be eased and relatively small matrices obtained if basis

states of the type ijj^(e) [l,m >, where the are only those

eigenfunctions of the electronic Hamiltonian that are

degenerate or nearly degenerate, are used. This procedure

will be employed to derive the eigenfunctions of the

coupled electron-nucleus quantum system in F e( Py N0)6(C1 0 4 ) 2

(Section 4.4).

The compounds studied in the present work will now

be examined with a view to determining whether the effective

field approximation of the magnetic hyperfine interaction is

2

-f-valid for the lowest electronic states of the Fe ion.

2

-f-The states of the ground doublet of the Fe ion

in Co Fe C l 2 are highly magnetically anisotropic (g = 0)

1 X X J_

and for this reason they are not mixed by the hyperfine

interaction (i.e. the hyperfine Hamiltonian within the

ground electronic doublet is diagonal). Thus, matrix

elements of the hyperfine Hamiltonian may be written in

the form:

< ^ i (e) |3C(e) I

\p±

><x.

where i[i^(e) represents electronic states of the type |a >

<a|JC(e)|si> is non-zero for some of the forms

of 3f(e) (e.g. L ,S ) but z z

<a 13C( e ) I b> is zero always. These matrix

elements ensure that the electronic and nuclear wavefunctions

may be determined separately since the hyperfine Hamiltonian

cannot mix these electronic states. In (M Fe )(PyNO)G~ l -x x

(0104)2? however, the magnitude of the splitting of a

similar ground electronic doublet by the presence of a

site distortion is of the same order as the magnetic

hyperfine interaction. The resultant non-magnetic rhombic

states, \fj i and

ip

« 1 2 - — (|a> ± |t> >)

’

Jl

The trigonal wave f unc t ions |3-> and |b> of Figure 2.1 (for

Fe(PyN0)6(C104)2) are of the form:

| a > 0 ._{9 9 4»! I} - 2 > - 0 . 1 0 4 » I - 1 >

+

_-©

-1 0 V

11

A

0 .104)! I

o> -

0 0 * 99 4)_ 1 1 2 >

w h e r e <j) , (J) a r e t h e w a v e f u n c t i o n s o f t h e g r o u n d o r b i t a l

t r i p l e t ( 5 T 2

_{) •}

I t i s c l e a r t h a t t h e r h o m b i c s t a t e s m a y

be mixed by the magnetic hyperfine interaction, i.e.:

| 3 C j ^ >

+

It follows that the effective field approximation is not

2 . 3 . 1 O r i g i n s of t h e M a g n e t i c H y p e r f i n e I n t e r a c t i o n

T h e r e a r e t h r e e m a j o r c o n t r i b u t i o n s to t h e m a g n e t i c

d i p o l e h y p e r f i n e i n t e r a c t i o n . T h e f i r s t a r i s e s f r o m t h e

i s o t r o p i c F e r m i c o n t a c t i n t e r a c t i o n p r o d u c e d b y t h e d i r e c t

o v e r l a p of u n p a i r e d e l e c t r o n s w i t h t h e n u c l e u s . T h e s e

e l e c t r o n s a r e m a i n l y i o n i c c o r e s e l e c t r o n s w h i c h h a v e a

n e t d e n s i t y at t h e n u c l e u s d u e to e x c h a n g e i n t e r a c t i o n s

w i t h t h e o p e n s h e l l e l e c t r o n s . T h e t w o o t h e r t e r m s r e s u l t

f r o m t h e o r b i t a l a n d s p i n m o m e n t s of t h e i o n . B o t h of t he

l a t t e r c o n t r i b u t i o n s a r e a n i s o t r o p i c a n d c a u s e t h e a n i s o t r o p y

w h i c h m a y b e o b s e r v e d in t h e m a g n e t i c h y p e r f i n e i n t e r a c t i o n .

In d e t a i l , t h e t h r e e c o n t r i b u t i o n s o r i g i n a t e as f o l l o w s :

(a) W h e n t he t o t a l o r b i t a l a n g u l a r m o m e n t u m L is

n o n - z e r o t h e r e is a c o u p l i n g w i t h t h e n u c l e u s a r i s i n g f r o m

t h e o r b i t a l m o t i o n of t h e o p e n s h e l l e l e c t r o n s ( t h e 3d

e l e c t r o n s f o r t h e f e r r o u s i o n c a s e ) . T h e o r b i t a l c o n t r i b u t i o n

to t h e m a g n e t i c h y p e r f i n e H a m i l t o n i a n m a y b e e x p r e s s e d

(A b r a g a m a n d B l e a n e y , 1 9 7 0 ) :

d C = 2 g 3 3 < r _ 3 >(L . I) 2. 14

L n n — —

w h e r e 3 is t h e B o h r m a g n e t o n a n d r is t h e 3d e l e c t r o n r a d i a l

c o o r d i n a t e .

(b) T h e c o n t r i b u t i o n f r o m t h e d i p o l e m o m e n t o f t h e

e l e c t r o n i c s p i n d i s t r i b u t i o n is g i v e n by:

( 3 S • r ) ( r • I )

2g 33 < r 3 >

n n - S . I 2 . 1 5

T h i s c o n t r i b u t i o n is n o n - z e r o o n l y w h e n t h e o r b i t a l s

r e l a t e d to the v a l e n c e e l e c t r i c field g r a d i e n t w h i c h r e sults from an a s p h e r i c a l charge d e n s i t y ( A b ragam and Bleaney,

1970). U s i n g e q u i v a l e n t o p e r a t o r s E q u a t i o n 2.15 may be w r i t t e n in the more c o n v e n i e n t form (Abragam and Bleaney, 19 70) :

K = - 2g ßß < r ~ 3> C [ | ( L - I ) ( L - S ) + 4 ( L - S ) ( L - I ) - L ( L + 1 ) (S .I )]

D n n z — — — — l — — — — — —

2 . 16

w h e r e £ is a co n s t a n t d e p e n d e n t upon the e l e c t r o n i c

1 2 +

c o n f i g u r a t i o n of the ion (- for Fe ).

(c) The c o u p l i n g b e t w e e n the n u c l e a r m a g n e t i c m o m e n t and the u n p a i r e d e l e c t r o n d e n s i t y at the nucleus,

the s o - c a l l e d Fermi c o n t a c t i n t e r a c t i o n , has the form ( A b r a g a m and Bleaney, 19 70) :

K

I

c 3 n n V i l —

l

w h e r e the delta f u n c t i o n r e p r e s e n t s the e l e c t r o n d e n s i t y at the nucleus. In t r a n s i t i o n m e t a l ions this i n t e r a c t i o n a r i s e s p r e d o m i n a n t l y b e c a u s e of p o l a r i s a t i o n of the inner s e l e c t r o n s by the 3d e l e c t r o n s . D i f f e r e n t e x c h a n g e

i n t e r a c t i o n s e x p e r i e n c e d by e l e c t r o n s of o p p o s i t e spin o r i e n t a t i o n p r o d u c e a r e s u l t a n t d i f f e r e n c e in d e n s i t y for s e l e c t r o n s at the n u c l e u s (viz. |ijj^(0)|2-|i|^(0)|2).

T h r o u g h the Fermi c o n t a c t i n t e r a c t i o n this net spin d e n s i t y at the n u c l e u s c o n t r i b u t e s to the h y p e r f i n e

Kc = - 2§n3 ß n <r 3> *(£•!) 2 . 1 8

w h e r e K is a n u m e r i c a l f a c t o r w h i c h m e a s u r e s t h e p o l a r i s a t i o n

To s u m m a r i s e , t h e m a g n e t i c d i p o l e h y p e r f i n e

i n t e r a c t i o n b e t w e e n a n u c l e u s a n d i t s s u r r o u n d i n g e l e c t r o n s

m a y b e e x p r e s s e d b y t h e H a m i l t o n i a n :

3C = 2g ßß I

m n n . —

-ri r i

3 ( r . •s .)(r . • JO

7 'I + --- 1--- —

---+ 6 ( r . ) ( s . • I )

3 l i — 2 . 1 9

w h e r e t h e i n d e x i r e f e r s to t h e e l e c t r o n s of t h e i o n .

C o n t r i b u t i o n s f r o m e l e c t r o n s in c l o s e d s h e l l s v a n i s h

l e a v i n g t e r m s f r o m o n l y t h e 3d e l e c t r o n s , e x c e p t of c o u r s e

f o r t h e d e l t a f u n c t i o n t e r m . E q u a t i o n 2 . 1 9 m a y b e r e w r i t t e n

u s i n g o p e r a t o r e q u i v a l e n t s as:

U = 2g ß ß <r 3> { L - I - C [ | ( L . I ) ( L - S ) + 4 ( L . S ) ( L . I ) - L ( L + 1 ) ( S . I ) ]

m n n — — 2 — — — — 2 — — — — — —

- K(S_.I_)} 2 . 2 0

%

It is p o s s i b l e to c o n s i d e r t h i s i n t e r a c t i o n , u n d e r

c e r t a i n c i r c u m s t a n c e s , in t e r m s of a m a g n e t i c f i e l d at t h e

n u c l e u s , p r o d u c e d b y t h e e l e c t r o n s , c o u p l i n g w i t h t h e n u c l e a r

m a g n e t i c m o m e n t - t h e e f f e c t i v e f i e l d a p p r o x i m a t i o n .

R e c a l l i n g t h e f o r m of E q u a t i o n 2 . 1 3 t h e e f f e c t i v e f i e l d ü e f£

m a y b e w r i t t e n :

H = - 2 3 < r _ 3 > { L - £ [ |l ( L . S ) + | ( L •S ) L - L ( L + l ) S ]

— eft — 2— — — i — — — —

The field is considered to result from three major

constituent f i e 1ds ■ (M a r s h a 11 and Johnson, 1962)

corresponding to the orbital, dipolar and Fermi contact

interactions. The relative magnitudes of the contributions

are determined by the electronic configuration and the

environment of the ion.

2 •4 Relative Intensities of Absorption Peaks

In the presence of a magnetic field and/or an

electric field gradient the nuclear states will, in general,

not be pure states (i.e. there will be mixing of nuclear

substates). This is evident from an examination of the

magnetic dipole and electric quadrupole Hamiltonians

(Equations 2.20 and 2.10) for systems in which the principal

axis of the e.f.g. is not along the magnetic field direction

and r) is non-zero. Transitions between these states for

57

Fe allow the observation of eight line Mossbauer spectra.

The intensity formalisms appropriate to magnetic dipole

transitions for single crystal and randomly packed

p o 1y c r y s t a 1 1 ine absorbers will be described in this section.

Reduction of the generalised expression to some simpler

commonly occurring situations will also be discussed.

2.4.1 Single Crystal Absorbers

The system under consideration is one in which

transitions occur between generalised states (of the type

shown in Equation 2.22) arising from a nuclear spin I = — e 2

and a spin I = ~ manifold, as for 57Fe, with I values of