A THESIS

ENTITLED

Magnetic,electrical and structural properties of

some La,Y and Sc based rare earth alloys

by

BEHZAD SHARIF

submitted for the Degree of Doctor of Philosophy

in the University of London

Imperial College of Science and Technology

The Blackett Laboratory

Prince Consort Road

London S;J7 2BZ

May 1979

I dedicate this thesis to all those Iranians who gave

their lives during the Revolution in 1979

1.

INTRO D UGT_I ON.

During the last 20 years the rare earth metals and alloys

have become the subject of intensive investigation. Starting

in the late 1950's Spedding, s. Legvold and their students

succeeded in growing pure single crystals and in measuring

their basic thermodynamic, magnetic and transport properties.

In 1960's W.C. Koehler and his colleagues at the Oak Ridge

National Laboratory and Hans Bjerrum MMller and his colleagues

at Ris0 research Establishment, using neutron techniques, were

able to obtain a rather complete experimental understanding of

the magnetic. interactions in the heavy rare earths. During 1970's

a number of neutron measurements were initiated on single crystals

of the light rare-earth at ^is% and on investigation of the

magnetic properties of the H.C.P metals and alloys of the heavy

rare earths (Gd-Yb) at Oak Ridge ( which led to a fairly good

understanding of their magnetic character).

Although the role of the neutron technique was crucial in

elucidating the magnetic properties of the rare earths, many

other measurements of, for example, magnetic susceptibility,

magnetostriction, heat capacity, transport properties, electro-

magnetic absorbtion, nuclear magnetic resonance and Mossbauer

effect have all made valuable contributions, while theoretical

interpretations and predictions have played a vital part in

suggesting new lines of investigations. Indeed the last twenty

years provides an excellent example of the interplay between

theory and experiment, and of the complementarity of different

experimantal techniques, which are so characteristic of modern

2.

solid state physics.

Since the original investigations of Klemm and Bommer in

1937 numerous investigations have served to complete the classi-

fication of the room temperature crystal structure. The most

significant contributions to an understanding of the structure

of rare earth metals has come from an examination of rare earth

alloying behaviour and from the observation of several polymor-

phic transitions induced by the application of high pressure td

the metallic elements.

There exists a trend in the sequence of crystallographic

transitions of the form h.c.p ---= Sm-type d.h.c.p

as a function of pressure, temperature and impurity concentra-

tion. This is the same sequence which exists across the rare

earth elements with decreasing atomic number. While this sequence

occurs in the direction f.c.c h.c.p for increasing density

in the case of pure metals ( low Z to high Z ), decreasing

temperature and the alloying of a light rare earth with increa-

sing concentration of a heavy rare earth, the reverse is true

for observations of the high pressure behaviour that is, with

increasing pressure and hence again increasing density, the

series is crossed in the reverse order h.c.p f.c.c:

These changes represent a remarkable set of experimental

data in which the same systematic structural changes occurs in

a family of metals which are closely related electronically as

function of three and possibly four separate variables.Whether

or not these parameters act in a similar way in producing the

* W. Klemm and H. Bommer, Z. anorg.U. allgem. Chem.,231,138(1937)

3.

phase transitions is not yet clear , although attempts have

been made at correlating the appearance of the different struc-

tures with a variety of physical properties of the metals.

4.

ABSTRACT

The interesting sequence of structural changes in light-

heavy rare earth alloys invites an investigation of the trans-

port magnetioproperties of this series of alloys. This thesis

is concerned primarily with the properties of these alloys as

well as certain of the heavy - heavy rare earth alloys.

In discussing the magnetic properties of rare earth (RE)

metals we may consider the partly filled 4f shells to have

essentially the same character as in RE3+ free ions but are

coupled via their interactions with and through the conduction

electrons. It is therefore appropriate to review briefly the

magnetic character of these 4f shells ( chapter 1 ) and then

to consider their interactions and how these interactions could

give rise to magnetic order ( chapter 2 ).

The tools used in the course of these investigations(a.c.

susceptibility and electrical resistivity) are described in

chapter 3.

In chapter 4 we present some of the experimental results

on the Y - RE and Sc - RE alloys ( where RE is a heavy rare

earth ). In dilute alloys the results provide good evidence

for the theoretical calculation of resistivity using s-f ex-

change .

Chapters 5 and 6 centre on the major concern of this thesis,

light heavy rare earth alloys. The magnetic character of these

alloys in complicated by the need to involve heavy rare earth

with ground state terms given by J=L+S in contrast to the J=L-S

terms of the light rare earth host.

5.

This situation can be simplified somewhat by the use of

Y as an (effectively) heavy rare earth and La as a light rare

earth as these elements do not sustain a magnetic moment.

Chapter 5 is concerned with the magnetic character of the Y-Nd alloy system. This chapter provides a rather complete set

of results concerning the magnetic properties of the Y-Nd alloy

system for the whole range of concentration and structure. In

this chapter it has been shown how theory could account for the

observed stability of the f.c.c. phase-field in this system.

Chapter 6 is concerned with the magnetic properties of the

La-Tb and La-Dy alloy systems. The observation of unexpected

anomalies in the resistivity and susceptibility of some of the

alloys in these systems has been attributed to the polycrys -

tallinity of the alloys: A more complete understanding of this

effect must await the measurement of other properties in addi-

tion to susceptibility and resistivity.

Finally in chapter 7 we have tried to understand the source

of differences in reported values of magnetic ordering tempera-

ture of Gd Al2.

6.

ACKNOWLEDGEMENT

The work presented in this thesis was carried out under

the supervision of professor B.R. Coles. It is my pleasure to

thank him for the many stimulating ideas, guiding influence and

for valuable discussions.

I would like to express my deepest gratitute to Dr. B.V.B.

Sarkissian who has not only taught me everything relating to

the practical side of this work but who has, throughout the

whole of the time, been passionately involved with this work

and been the source of many fruitful discussions.

I thank Dr. H.E.N Stone for his involuable advice on all

metallurgical matters related to this work and I specially

thank him very much for showing great patience in dealing with

me and my broken English during the early stages of this work.

I also thank him for his great willingness to always be of

assistance. I would also like to thank all the other members

of metal physics.

I am greatly indebted to my fiancee Miss Sh. Zand for

patiently typing this thesis without any previous experience

and for her continuous encouragement and moral support during

the period of this work.

I acknowledge the financial support of the Atomic Energy

Organization of Iran during most of the time spent at Imperial

College.

CONTENTS

Introduction

Abstract

Acknowledgement

•Contents

Page

1

4

6

7 Chapter 1 Magnetism in Metals

Introduction 11

1.1 Diamagnetism 11

1.2 Paramagnetism 14

1.2.1 Paramagnetic susceptibility 26

1.2.2 Paramagnetism in metals 31 1.3 Ferromagnetism 38

1.3.1 The exchange interaction 42

1.3.2 Spin waves 44

1.3.3 Band model of ferromagnetism 46

1.3.4 Crystalline anisotropy 47

1.4 Antif erromagnet ism 48

1.4.1 The molecular field model of antiferro -

magnetism 49

1.5 The demagnetization factor D 51

References 52

Chapter 2

2.1

2.2

Rare earth metals

Structure behaviour of rare earth metals

and alloys

Magnetic properties

a) Spin contribution

54

57

7

8.

Page 2.2.1 The indirect exchange interaction or

R.K.K.Y interaction

b) Orbital contribution

2.2.2.1 The crystal field magnetism 62

2.2.2.2 Magnetostriction and elastic energy 72

2.2.3 Magnetic ordering 76

Thermal first order transition from spiral

to ferromagnetic arrangement 86

2.3 Transport properties ( electrical resis —

tivity) 99

2.3.1 Spin disorder resistivity 102

2.3.2 Spin wave scattering 104

2.3.3 The effect of suuerzone boundaries 105

2.3.4 Crystal field effects 108

2.3.5 The effect of alloying

a) Dilute alloys 109

2.3.5.1 Kondo effects 111

2.3.5.2 Crystal field effect 111

b) More concentrated alloys

2.3.5.3 Spin glasses 113

References 117

Chapter 3

3.1

Experimental methods

A.C. susceptibility apparatus 124

Multiturn test mutual inductance 125

Cryostat and thermometry 127

The diode thermometer 128

Calibration of diode thermometer 131

9.

Page 3.2 Electrical resistivity apparatus 131

Cryostat and thermometry 131

Thermometry 134

The carbon resistance thermometer 135

The thermocouple 136

Pt resistance thermometer 136

Electrical circuit 137

3.3 Experimental procedure 138 3.4 Specimen preparation 139

References 141

Chapter 4 Results and discussion of Sc-RE and Y-RE

solid solutions ( RE : Er, Ho and Dy ).

Introduction 142

4.1 Dilute alloys 142

4.2.1 Sc-Er alloys 146

4.2.2 Y-Er alloys 153 4.2.3 Y-Ho alloys 153 4.2.4 Y-Dy alloys 160

References 163

Chapter 5 The magnetic character of the stable and metastable phases in the neodymium -

yttrium alloy system

Introduction 164

5.1 Solid solution in yttrium 165

5.2 Alloys containing the samarium structure

phase 174

5.3.1

5.3.2

10.

Fase D.h.c.p alloys 176

The f.c.c alloys 19z4-

Discussion 199

Conclusion 200

References 201

Chapter 6 Magnetic and electrical character of the

La-Tb and La-Dy alloy system

Introduction 202

6.1 La-Tb system 202

6.2 La-Dy system 217

Discussion 221

References 229

Chapter 7 In search of the sources of difference in

reported values of Tc in Gd Ale . Introduction 230

Results and discussion 231

Conclusion 235

References 236

CHAPTER 1

MAGNETISM IN METALS

Introduction

1.1 Diamagnetism

1.2 Paramagnetism

1.2.1 Paramagnetic susceptibility

1.2.2 Paramagnetism in metals

1.3 Ferromagnetism

1.3.1 The exchange interaction

1.3.2 Spin waves

1.3.3 Band model of ferromagnetism

1.3.4 Crystalline anisotropy

1.4 Antiferromagnetism

1.4.1 The molecular field model of antiferromagnetism

1.5 The demagnetization factor D

References

INTRODUCTION

There are five classes of magnetic materials namely,

ferromagnetic, ferromagnetic, antiferromagnetic,

paramagnetic and diamagnetic. Diamagnetism is common to

all matter, but is very weak. Ferromagnets, ferrimagnets

and antiferromagnets become paramagnets at sufficiently

hightemperature.

We define magnetic susceptibility )(=' .fi , where

N is magnetic moment per unit valume and H is magnetic

field intensity is negative for diamagnets, and positive

for other materials. Atomic theory shows that the magnetic

dipole moment observed in matter arise from the orbital

motion and spin of electrons. A small contribution also

arises because of the nuclear magnetic moment.

1.1 DIAMAGNETISM

Diamagnetism is a perturbation of the orbital motion

of electrons of a character to oppose a flux increase

through orbital loops. It is common to all substances,

even in metals where there is also a contribution from

conduction electrons as well as from the ion cores.

Consider an electron in an orbit. If we apply a

magnetic field H ,the change in magnetic moment of the

electrons Dr classically is :

L~ = e2x a2% 6m where < a2) is the mean square distance of an electron

11.

12.

from the nucleus.

Now for N atoms per unit volume, and for Z electrons

per atom the susceptibility is given by :

X = - NZe2 / a2 \ 6mH ` j This is the classical Langevin formula for diamagnetic

susceptinility of atoms. The same formula is obtained in

quantum theory, in which simple orbital motion is no

longer envisaged. < a2>now depends on the charge distribution

and the problem of calculating the susceptibility is really

a problem of calculating(a2)from quantum mechanics. The

calculations are difficult if many electrons per atom are

present, and approximations must be employed.

The Langevin formula suggests that the diamagnetic

susceptibility should be temperature independsant apart

from changes in Niue to lattice changes. This is verified by

experiment.

Consider now diamagnetism of an electron gas. That is

the orbital motion of the conduction electrons in a metal

( such as copper ) under the influence of the magnetic

field, H, the electrons will undergo a helical motion.

The translational motion along the field direction will be

unchanged, and we can ignore it. The projection of the

motion onto a plane prependicular to H is a circle. By

equating the magnetic force evB to the centripetal force,

the angular frequency of the circular motion is found to be

eB (m ks units). The orbital motion sets up a dipole

moment which is in the oposite direction to B.

Remarkably enough when collisions are considered and

13•

classical statistics are employed the diamagnetic suscep-

tibility due to the conduction electrons is found to be

zero. However electrons in a metal obey Fermi-Dirac

statistics and the available electron energy states are

quantized. ( Quantum -mechanically an electron in a circular Vd

orbit in a magnetic field is equ3lant to two harmonic

oscillators each __quantized.) We find that the electron

energies are given by

E _ ( n + )h e B/ 2mA Yhere, n = 0, 1, 2, . These levels are called

LevcLs Landau,. They are formed by bunching of the-usual levels

( B = 0 ) of electrons in a 3 dimensional box. The Landau

levels are therefore highly degenerate. The diamagnetic

susceptibility of the conduction electrons is determined

by calculating the thermal distribution over the possible

energy states. Using Fermi-Dirac statistics, as we should

for electrons, we get a negative susceptibility independent

of temperature• (apart from changes due to N,)

)(7--7 — - ( pli )'11B2 ( 1.1 )

3h where ig -- - is the Bohr magneton. As we shall see later,

the paramagnetic susceptibility due to the spin of the

conduction electrons is three times as large as the above,

and of opposite sign.

In the above equation we assumed the free electron

model; but the motion of an electron will actually be

perturbed by the periodic potential of the core. A useful

approximation to account for this is to replace the mass

m by an effective mass m .

14.

The total susceptibility of a simple metal is the sum

of three terms :

1 - Susceptibility of the core atoms.

2 - Susceptibility of the orbital motion of the conduction

electrons.

5 - Paramagnetic susceptibility of the spin motion of the electrons.

The first two contributions are negative, while the

third is positive. Depending on their relative magnitude,

the total magnetic susceptibility of a simple metal may

be positive or negative. It will defend on the number of

conduction electrons per atom, and the number of bound

electrons per atom.

1.2 PARAMAGNETISM

Let us review some quantum -mechanical and spectroscopic res-

ults.Application of Schrodinger's equation, to atoms leads to

4 quantum numbers : n, 1, ml, and ms (and also spin quantum

number s = -).

n = 1, 2, denoted by k, 10-6,

1 = 0, 1, 2, ( n - 1 ) denoted by s, p, d,

The orbital angular momentum is given by 'hi(i+1)1 .

Ypl determines the projection of this onto any axis

( usually the magnetic field ) . 1 '! has the values + 1, (1-1) , -1, and the associated angular momentum isIntii•ms has the

values ± , and ►'lgii is the projection of the total spin

angular momentum of an electron onto an axis. The total

15.

spin angular momentum of an electron is }ls. ( s + 1 )

where s = .- always. The relation between the angular

momentum L and the magnetic moment hl is : e

2m-

The components of pL are .: m

ml$ or ml B , where

realize that the electron spin has no classical analogue

( except for the fictitious model of a spinning sphere of

charge ), and it is a consequence of relativisitic wave

mechanics, as first shown by Dirac in 1928. In this

relativistic quantum theory the electron spin having the

observed angular momentum and magnetic moment emerges in

a natural way with the three quantum numbers n, 1, ml.

The spin angular momentum is given by :

S.- _ i[s (s + 1)) 4;

where s=- always. The component of S along any axis are +A

However it turns out that the magnetic moment associated by

the sain angular momentum is not given by em I SS , but this

must be multiplied by a factor g, the spectroscopic splitting

factor, so that the spin magnetic moment is :

s = g ( 7E—) s = g () ( 3 ) 7/2

and the component of this along an axis ( usually the applied

magnetic field ) is g ( ) fih-- for a free electron,

g = 2.0023.

In a free 'atom, there are two contributions to the total

angular momentum ( and hence magnetic moment ) : the orbital

angular: momenta of the electrons, and the spin angular

B = e 2m

Consider now the electron spin. It is important to

16.

momenta. In the vector model of the atom we introduce an

additional quantum number J, which determined the total

angular momentum due to vectorial addition of the orbital

and spin angular momentum. Thus for a single electron J

is always half-integral, being 1 ± 4.

It is now necessary to consider how the orbital and

spin momenta of the electrons of an atom combine to form

the atom's total angular momentum. The method of combination

that is important in magnetism is known as Russell-Saunders

couplinP). The L vectors of various electrons are added

( vectorially) to form a resultant L , whereas the s-vectors

are assumed to form a resultant S.

The resultants S and L are then combined ( vectorially)

to form the total atomic angular momentum. The associated

quantum number is J.J can take the range of values J=(L-Sr)',

( L - S + 1 ), ( L + S ) (1.2)

and'such a group of levels is termed a " multiplet ". By

definition, the multiplicity of the system is 2s + 1 (i.e,

there are 2s + 1 values of J). This multiplicity is only

developed if L is greater than, or equal to S. If L is less

than S, there are only 2L + 1 values of J. Because of spin-

orbit coupling, different values of the multiplet ( which

correspond to different values of J ) do not have the same

energy. The spacing of the levels is determined by the spin-

orbit coupling constant \, defined so that the interaction

energy is given by .,/\ Z . S . The interaction between the

orbital magnetic moment, and the spin magnetic moment of an

electron can be understood in the following way : If an

17.

electron is orbiting the nucleus, then an observer fixed

with respect to the electron ( but not spinning with it)

sees the nucleus orbiting it.The orbiting positive charge

produces at the site of the electron a magnetic field

whose magnitude and direction depends on the magnitude and

direction of the electron's orbital angular momentum. This

magnetic field acts on the spin magnetic moment. The

electron's energy will depend on the orientation of the

spin magnetic moment in the magnetic field. The spin-orbit

interaction exists in all orbital states except S states

( where the orbital quantum number L = 0 ).

The spin orbit interaction energy is A Z.S

1ISl cos e where g is the angle between L and S .

But L = hL. 1 C 1+1 ),

and similarly

S = h Cs ( s + 1 ),-

and since -5 -4.. -5 J = L + S

we get :

• AL.s = [J( J + 1) -L( L+ 1)-S (s + 1)J h2

is a constant for a given multiplet, but may be

different for different multiplets. The total angular

momentum for the atom or ion is given by i L J (J + 1)1

In the presence of a magnetic or electric field which is

not strong enough to break up the coupling between L and S,

the energy of the atom is quantized into 2J + 1 levels.

The contributions of the orbital and spin angular momenta

18.

to the total magnetic moment are different ( the spin

angular momentum gives twice the contribution per unit of

angular momentum as does the orbital angular momentum ).

We can define a total effective . g- factor by writing the

total magnetic moment as :

eli 7 geff [J(J + 1)]

where geff reffers to the total atom or ion. geff can be

found by carefully considering the various vectors involved

in Fig:1..1(2)

geff=1 +JJ +1)

2Js((s++1}) -1 (1+1) (1.3)

The vectors L and S are added to give the vector J, and the

magnetic moment vectors are added to give a resultant f4,Fig.1.1.

Note that the vector ti is not in general along the same

line as the resultant angular momentum vector. In fact '1,1 SI will

process about the axis of the resultant angular momentum

vector and the average magnetic moment component prependicular

to this axis will be zero.

The above result has been obtained by employing vector

model of the atom. Quantum mechanics gives the same results.

Note that if S = 0 then g = 1, and L = 0, g = 2, as expected.

The ground state of an atom or ion corresponds to one

of the levels of the multiplet previously ref..ered to.

The ground state of an atom or ion and any other states

are described by certain values of J, S, and L. The spectros-

copic notation for a state is defined by taking S, P,

to denote the value of L, with a prefix denoting 2s + 1, and

a suffix denoting J. For example if L = 2, S = 2, and J = 4,

the state would be described by, 5D4, and in general the

r,

IA J

19.

20.

state is described by :

2s + 1 L J

where L is S, P, D, F denoting the values

0, 1, 2, 3

It remains now to determine how the individual L and S

vectors combine to give L and S and then to determine how

the vectors L and S combine to give J for the ground state.

From studies on spectra, Hund arrived at three rules that

permit the prediction of the magnetic moment of free atoms,

or ions, in their ground states. These rules are :

1 ) S = msi principle .

2 ) L =Emli

is the maximum allowed by the exclusion

is maximum ( after S has been maximized)

3 ) J for an incompletely filled shell is given by :

J = L-S for a shell less than half filled

J = L+S for a shell more than half filled

As examples of the application of Hund'S rules consider the

following rare earth ions in the free state. ( As we shall

see later, ions in crystal are affected by their neighbours

and the orbital angular momentum may not be that for an ion

in the free state).

Dy3+ has 9 electrons in the 4F shell ( as well as the two

electrons removed from 6s, the third electron is removed

from the 5d shell the same happens for the other rare earth).

m = 3 2 1 0 —1 —2 —3

IV ti It t 1

Each arrow represents an electron spin. Thus s=5/2 and

21.

L = 5 . Since the shell is more than half-filled,J =L+S =15/2.

Hence the ground state of the free D 3 ion is 6H15/2. SIA+, and Eu3+ have 6 electrons in 4f

S =3 , L = 3 and J = L - S = 0 and we have

7F0 for the ground state. (.These free ions, with J = 0, thus

would be expected to have zero magnetic moment ).

Hundr's 1st. rule is an indication of the fact that the

exchange interaction between electron spins favors a parallel

alignment. This arises from a combination of the exclusion

principle and the coulomb interaction. Two electrons whose

spins are parallel cannot share a common small volume of

space ( although they can if the spins are antiparallel).

Hence the coulomb energy is minimized for parallel spins.

Hund's 1st. and 2nd. rules give the L and S values for the

ground state of an ion. The spin-orbit coupling determines

how L and S vectorially combine to give J, and Hund's third

rule tells us which combination has lowest energy. Different

combinations of L and S form states in a multiplet described

earlier.

A multiplet of term will be formed for each possible

set of L and S. That is, the individual orbital and spin

angular momentum may add to give different values of Land

S. Different terms will have different energies. That is

while Hund's rules gives L and S for the ground state.

excited states will have different L and S.

For example consider the splitting of the various levels

of an electron configuration, 4P1 4d1. Here L = 1 for the

22.

the .P electron and L=2 for the d electron.

Therefore the totalL can be „ 2,or 1. Independently the

spins can add to give S = 0 or S = 1. The fine-structure

splitting is shown in Fig 1.2(2)

tp 'Da E

4P '9d'

z 0 3"

3 %

S=1 / 3-D F

unperturbed spin-spin residual spin-orbit

state exchange energy electrostatic energy energy

Fig4.2

The ground state is 3F2, which is given by Hund's rule.

This state has J = 2, and can be further split by a magnetic

field into the M = 2, 1, 01-1,-2 levels.

The atoms or ions which generate magnetism in solids

are usually " transition elements ". These elements occur

in the periodic table when electrons enter an outer S shell

23.

before completely filling inner d or f shells, like the

rare earth group ( unfilled 4f shell ) .

So far we have only considered free ions. When a

magnetic ion is placed in a solid it is acted on by the

electrostatic fields of its diamagnetic neighbours. That is

an ion experiences an electric field due to its neighbours

and this field acts on the orbital motion of the electrons

of the magnetic ion. The free ion will have a total L which

will combine with the total S to give a total J. The ground

state will be 2J + 'I fold degenerate in the absence of an

electric or magnetic field. But we have an electric field

such as that produced by the neighbours of an ion in a

crystal, then the degeneracy of the ground state may be

removed. This can occur in three ways :

1 - If effect of electric field is small, then L and S are

still coupled to give J, and the ground state will split

into the various MJ components. This happens in the rare

earths, where the 4f shell is partly shielded from the

crystal fields by the outer 5S and 5d shells.

2 - The effect of the electric field can be strong enough

to break the coupling between vectors L and S, and then

each vector will precess independently about the electric

field direction. J then has little meaning. The electric

field splits the various ML levels, each (2s + 1) fold

degenerate in spin. This degeneracy may also be partly

lifted by crystalline field.

3 - If the electric field is very strong, the coupling between the orbital angular momentom of the individual

24.

electrons, and between the spin angular momenta of the

individual electrons, may be broken. This case usually

corresponds to covalent bonding.

Let us return to case 2. This situation occurs in

the iron group transition elements, where the unfiLCed 3d

shell is more exposed to the crystal fields than, say, the

4f shell of the rare earth group. The splitting between

various ML levels for ions of the iron group is usually

about 104 Cm1, and hence only the lowest CLQ level is

populated at ordinary temperatures. ( KT"' Cm1 at room temp.) U Ii

The orbital motion is then said to be quenched since it

averages to zero. Physically, we can imagine that an electron

moving in an orbit in an inhomogeneous electric field has

its orbital direction continously changed and the time

average of the orbital angular momentum ( and hence magnetic

moment ) is zero. We can look at this phenomenon another

way : If only the lowest orbital level is occupied, an

applied magnetic field can not affect the- distribution of the

electrons over the orbital states, and hence the orbital

contribution to the magnetic moment in the direction of the

magnetic field is zero. The spin states are little affected

by the field, and only the distribution over the Ms levels

is changed.

The effective magnetic moment of the ions of the iron

transition group, derived experimentally, is closer to

g ~tB (s(s + 1) )~ (where g = 2 ) rather than g413 (J(J+1) )7

(where g is given by equation(1 .3)Lande formula) . This

indicates that the orbital contribution is quenched.

25.

Let us see the physical reason for splitting of the

levels in crystalline field. As an example suppose that

we have an ion with L = 1, so that m1=1 , 0, -1. For each

of these values of mL there corresponds a certain

probability distribution for the electrons in real space•or

effectively different charge distributions. Consider the

three possible degenerate charge distributions shown in

Fig.1.3 for a free ion with L = 1.

a) b) c)

Fig. 1.5

Now suppose the ion is placed in the centre of an 11 It

octahedral enviroment shown in Fig. 1.3. a.

Fig. 1.3.a.

26.

Six diamagnetic neighbours are at the corners of the

octahedron. These are surrounded by their electrons of course

that is by their electron. clouds. Since electron clouds tend

to repel each other,the electron cloud b) and c) above will

be arrangements with lower energy than a). Hence the

degeneracy of the distributions have been lifted and the

orbital levels may look like Fig. 1.4.

LZ

Energy of ion

L = 1

Free ion Crystal field splitting

Fig. 1.4

1.2.1 11

. PARAMAGNETIC SUSCEPTIBILITY tl

In 1895 Curie showed that certain substances did not

have a. temperature independent susceptibility, as expected

for a diamagnet, but rather the magnetic susceptihility

was given by :

=C / T

Where T is the abE3olute temperature, and C is a constant

called Curie constant which depends on the substance.

27.

The susceptibility here is positive and larger in magnitude

than the diamagnetic susceptibility by a factor of 10-2 or103 Later experiments showed that there were many compounds

whose magnetic susceptibility could be described more

accurately by the relation :

Known as Curie-Weiss law. 6 is a constant which can be

negative or positive.

Materials which obey the Curie-Weiss law are called

paramagnets.

So far, we have considered the origin of permanent

magnetic moments of free ions. Now we can derive the

experimental facts ( Curie's law, and Curie Weiss law ) by

considering the microscopic behaviour of the elementary

moments((. Assuming noninteracting elementary dipoles

Langeviāl)has derived the following expression for magnet-

isation and susceptibility ( assuming VA KT ) :

M= 2. 1

3KT 3KT ( 1.4 )

Where N is the number of the ions, FL is their magnetic moment.

Now let us consider the multiplet effect on suscepti-

bilit44)

a ) Multiplet splitting much greater than KT.

Here we assume that all the magnetic atoms are in their

ground state ( as given by Hund's rules ), and this ground

28.

state is characterised by the quantum number J. In an

applied magnetic field the ground state will split into the

various MJ levels, where

J, J-1, , -J.

The different MJ do not have the same energy in an applied

field, because each M state corresponds to a different

orientation (U =-N.H = -MJ.gpt H ) .Now if AH

29.

level, that is,(2J + 1). Then we get :

tS1 11 r {g J [g2J(J+1) /3KTJ -O(J) (2J+1 X-

N J -SI ` _

E(2J+1) e-E(J) /KT

-E(J) /KT

(1.6)

Here subscript J have been attached to g to show explicity

that it is a function of J. a(J) is a term arising because

of the effect of magnetic field on the energy levels of an

atom(4).

C) Narrow multiplets with respect to-KT.

If the multiplet spacing is small compared with KT,

the result is(4):

X N f2 3KT [LL+1)+sc+1)1

where L and S define the multiplet.

So far we have ignored any interactions between the

magnetic moments of ions. Later we shall see that these

interactions are very important in co-operative magnetic

phenomena. Without them,there would be no ferro-or antiferro-

magnetism, although we would have paramagnetism. It is the

interactions between the magnetic ions which give rise to

the Curie-Weiss law. Weiss first introduced the concept of

a molecular field acting on each ion due to its neighbours

that is, in addition to the applied field H, there is an

effective field due to the magnetic neighbours of an ion.

If we assume that this effective field is proportional to

the magnetisation of the neighbouring dipoles, then instead

30.

of H we have H + AM where X is the molecular-field coefficient. If we replace H by H + M the Curie law will be modified as follows :

X M C H T (1.7)

where H=

Happ + ~M

C(Hālp+7M)

T

C T-9

where Q = AC

Magnetic susceptibility measurements enable tk and e to be et found, and compared with theory. A theorical value ofe may

not be easy to obtain, however we can compare the experiment -

ally deduced IL with that calculated for the free ion using Hund's rules. Usually is found in units of Bohr magnetons,

and then compared with :

P = g(J(J+1)] Tablet:1 shows the theoretical and experimental data for

the susceptibilities of rare earth ions. The susceptibility

per atom is given by ~a p2/3IcT(5) .

so

M =

or

31.

The table indicates good agreement between theory and

experiment, except for Sia and Eu}3. For these two ions it

is found that the spacing of the multiplet levels is not

large compared with KT(as we assumed in derivation of the

Curie law),If this is taken into account and equation(1.6)

is used the results are in good agreement with the - experi!-

mental values (as shown in parentheses in table 1.1).

1.2.2 PARAMAGNETISM IN METALS

So far we have been considering the paramagnetism of

ionic crystals in which the magnetic ions have localized

magnetic moments at fixed points w°.ithin the crystal.

However, if a material is a conductor we have free electr-

ons in the metal and since each electron has an intrinsic

spin magnetic moment we must consider the possibility of

this electron gas being magnetised this will occure wheth-

er or not the core ions are themselves magnetic.

It is clear that the conduction electron system is

not just a system of free electrons in a metal.However we -ion

assume in our treatment that we do have a system of free conduct--

electrons , and hope that our results will give a reason-

able description of relatively simple metals such as Na,Cu

,Ag etc.Later we shall discuss the transition elements (in

elemental form) .

"Paramagnetism of an electron gas

For simple metals,we are concerned with the paramagn-

etism of the electron gas, i.e., with the magnetisation

Ion Ground

La3+ 1S Ce3+ 2F°

r3+ 3H4/2 P iv"d3+ 519/2 Pm I4 sm °x 3+ 3+ 5/2 Eu 7Fo

Gda+ 8S7/2

Tb3+ 7F7 Dy3+

6H15/2 Ho I8 Er3+

4115/2 Tm H

m3+ 2F7/2

32.

S L J gJ p2= g2 (J4.1) P2expt..

0 0 0 0 ( 0

1/2 3 5/2 6/7 6.43 6

1 5 4 4/5 12.8 12

3/2 6 9/2 8/11 13.1 12 2 6 4 3/5 7.2 -

5/2 5 5/2 2/7 0.71(2.5) 2.4

3 3 0 0(12) 12.6

7/2 0 7/2 2 63 63

3 3 6 3/2 94.5 92

5/2 5 15/2 4/3 113 110 2 6 8 5/4 112 110

3/2 6 15/2 6/5 92 90

1 5 6 7/6 57 52

1/2 3 7/2 8/7 20.6 19

Table 1.1 Theoretical and experimental data for the suscepti-

bilities of rare earth ions. The susceptibility per atom is

given by 1.1BP2

/3kT. The values in parentheses for Sm and Eu are calculated using the Van Vleck formula, equation(1.6).

33.

that might result from an alignment of the intrinsic

spin magnetic moments of the conduction electrons. We

have already considered the diamagnetism of the conduction

electrons resulting from the orbital or circular motion

of free electrons in an applied magnetic field in

section 1.1-In addition to the susceptibility arising

from this orbital motion,we must now consider the suscep-

tibility arising from the spin motion of the conduction

electrons.

If electrons obeyed Boltzman statistics then we

would expect the susceptibility due to the conduction

electrons to be given by

N R ) S (S+1) - 3KT

where N is the number of conduction electrons per unit

volume and S= . The result gives X# 10-4cm-3 for

T= 300K . However, experimentally we find that the observ-

ed susceptibility of metals(such as copper) is smaller than

this by a factor of perhaps 100. Also observed susceptibi,-

lity is only slightly temperature-dependent,and not pro-

portional to 1/T . The discrepancy is removed by the application of

Fermi-Dirac statistic6) Those electrons which are well

below the Fermi level do not change their orientation when

a field is applied. That is,the distribution of such elec-

trons over the available states is unchanged. However

these electrons nearer the Fermi energy and above can

34.

change their state, and if their distribution over the

available energy states is altered, there is a contribution

to the susceptibility given 147?

X = 3N r2/2KTF assuming T

35.

In the derivation of 1.8 all interactions have been

neglected. It is also assumed that EF is independent of

temperature, while this is not exactly true, it is a good

approximation. The interaction between electrons is an

exchange interaction, tending to keep apart electrons with

the same spin component. The interactions of the electrons

with the core ions also affects the susceptibility of the (1)

conduction electrons.

experimentally, the observed values of susceptibility

are given by :

)(total =)( Dia +X Dia +X para core core electrons conduction electrons

( assuming )(para-core = 0 ). These susceptibilities are all

small and somehow need to be separated to give a proper

comparision with theory. Generally, experimental values of

susceptibility are only in fair agreement with the above

theory, although they are of the right order of magnitude.

Indluding electron-core and: exchange - interactions can improve

the agreement ( although as experimental susceptibilities

are small and may have relatively large errors ).

Before proceeding to the next section we emphasize the

fact that in the derivation of Langevin formula ( equation 1.4)

it was assumed that, 6 N`KKKT to complete our discussion of

the paramagnetism we wish to remove the above restriction.

Assuming multiplet splitting large compared to KT we get the

following equation for magnetisation(1) :

(2J+1) (2J+1) ' - 47-)Ng~J

36.

where Y = Jg g H/KT

we can write

M = Ng µBJ BJ (Y)

Where B(Y) is called the Brillouin function(8). If H/T is

large Coth ((2J+1) /2j) Y and Coth Y/2J both approach unity,

so that N approaches Ng t8J the maximum component of magnetic

moment in the direction of the applied magnetic field.

- The approach to saturation was first observed(9)for

hydrated gadolinium sulfate, Gd2 (SO4)3. 8H20. This is parti-

cularly favorable case, since the Gd+3ion has L = 0, and

therefore no complications caused by the crystalline field.

It is also well diluted magnetically because of the eight

water molocules. Further work at higher values of field has

been carried out by Henry(10)on potassium chromium alum

K Cr(SO4)2.12H2O and ferric ammonium alum Fe NH4(SO4)2.12H2O

as well as hydrated gadolinum sulfate. Since L is quenched

for iron group ions, the value of S should be substituted

for J in the Brillouin function. The Brillouin functions

for J = 7/2 (Gd+3), together with Henry's result are shown

in Fig. 1.5. Note that if J-°° ,that is, when all orientations

of the moments become possible when a field is applied, we get

BJ (Y)= CothY - 1 = L(Y) Y

Where L(Y) is the Langevin function, which was derived by

assuming that all orientations of the dipoles are possible.

6

3

t ~

10 20 30

-k/T X 103 `de/deg Ī

Fig. 1.5. Saturation effects in high field and at

low temperature for various paramagnetic ions. (After

W. E. Henry.).

37.

38.

1 .3 . FERROMAIGNET IBM

We mentioned earlier that the parameter 8 in the Curie-

';ieiss law arises because of the interaction between atomic

dipoles. This interaction means that the total energy of a

pair of magnetic dipoles depends on their orientations with

respect to each other. We can define a ferromagnet as a

material-in which the elementary dipoles tend to align

parallel to each other.

In a ferromagnet, there may be a spontaneous magnetisation

of the atomic dipoles in the absence of all applied field. We 11 i1

may then have the familiar bar magnet . The tendency for

complete alignment of the dipoles is opposed by the magnetos-

tatic energy, which essentially is the energy recuited to set

up the magnetic field surrounding a magnet ( remember that a

magnetic field has an energy density B2/2Ibv ). It may be

energetically favorable for the material to form domains,

which are small regions with a particular orientation of

magnetic moment 11)

In a ferromagnet, the spontaneous magnetisation is

opposed by thermal energy, and as the temperature rises it

eventually reaches Te l the Curie temperature, above which

there is no spontaneous magnetisation, and the susceptibility

follows a Curie Weiss law.

Most magnetically-concentrated materials are antiferro-

magnetic or ferrimagnet, only a relatively small number are

ferromagnets. Of those materials which are ferromagnetic,

the majority are metals or metallic alloys. The few ionic

39.

ferromagnets include CrBr3,Eu 0, Eu S, EuSe,EuIl and Buy_ S i0. In contrast to this, there are more than 100 known ionic

antiferromagnets.

The elements Ni, Fe, Co, Gd are ferromagnets, while

Tb, Dy, Ho, Er and Tm are ferromagnetic at low temperatures,

up to some temperature Tc, and antiferromagnetic above that

up to TN,. where TN is the Nel temperature.

T© treat ferromagnetism, we consider the molecular

field model, developed by Weiss, in which the atomic dipoles

are presumed to experience an effective magnetic field (the

molecular field) proportional to the magnetisation of the

surrounding dipoles. We can make an elementary estimate of

the order of magnitude of the molecular field by comparing

the energy of a dipole in this field with thermal energy KTG.

Presumably KT is of the order of the molecular field, since

we require a temperature greater than T in order to destroy

the molecular field. If-we take an elementary dipole moment

rBthen,

where Bm is the molecular field. If T' —lo3 (e.g. Iron); then

Bm~107 gauss^'103W/m2

This field is much greater than the fields that can be produced

in the laboratory, it is also much greater than the dipolar

field expected to be produced by a neighbouring dipole.

(../1000 gauss.) • n

Thus the dipolar field is far too small to account for

40.

the observed effective fields in magnetic materials, and

Weiss was not able at the time (1907) to explain the magnitude

of the molecular field. In fact the origin of Bm is the

exchange interaction a quantom mechanical effect dependent

on the overlap of the atomic orbitals. ( A complete discussion

of exchange interaction is given in next section and chapter 2)

Application of molecular field yields(12):

To( =ein Curie-Weiss law) _ g 21,1J( J+1)

3K (1.9)

where Tis the molecular-field coefficient ( a dimensionless

quantity) and ; M(T) = B ( 3J Tc I1(T) 11(0) J 3+1 T M(0) )

Where M(0) is the spontaneous magnetisation at T = 0 and I1(T)

is the spontaneous magnetisation at temperature T, and BJ is

Brillouin function;8)introduced in previous section.

For a given value of J, the plot of N(T)/M(0) versus 1c

yields a universal curve•the experimental measurements for

Gd is plotted in Fig. 1.6, together with the Weiss theoretical

curve ( equation 1.10)(12?

as Fil 1.6

(1 .10)

1.0

41.

The magnetisation M(T) that we have discussed above is

not the actual magnetisation for a specimen unless the

specimen is a single domain. Instead, M(T) is the magnetisation

within a domain.

If the experimental value for M(T) fitted to Brillouin

function ( equation 1.10 ) we get the value of M(0) , the

magnetisation at T = 0. If the number of magnetic atoms per

unit volume is known, we can estimate the effective component

of magnetic moment per atom neff'

neff N M(0)

or

neff (in Bohr magneton)

( All magnetic ions are aligned at T = 0, that is, each has a

maximum component of magnetic moment in the same direction as

other magnetic ions). If the ions were simply localized with

magnetic moment gr,B (J(J+1)) , the maximum component would be

neff= g3. Some results for several ferromagnetic rare earth

metals are shown in table1.2.

Element Tc (K) neff(1418)

Gd 293 7.55 7.0

Tb 218 9.25 9.0

Dy 85 10.2 10.0

,gJ

Tablel•2

42.

In the last column, gJ is computed for the rare earths,

assuming free triply-ionized ions. Thus for the rare- earth

metals it seems that the magnetic moment arises from well

localized electrons in 4f states, as for free ions.

1.3.1 TEE EXCHANGE INTERACTION

We have seen that the magnetic dipole - dipole interaction

is too weak to account for the coupling between magnetic ions.

In 1928, Heisenberg -showed that the effective field is n tI

a result of a quantum mechanical exchange interaction .

The exchange field gives an approximate representation

of the quantum mechanical exchange interaction. On certain

assuption it can be shown(l)that the energy of interaction of

atoms i,j bearing spins Si, Sj contains a term :

E = - J. Si. Sj

Where J is the exchange integral and is related to the overlap

of the charge distributions of the atoms i,j. Equation(1.11)

is called Heisenberg model.

The exchange energy is electro-static in origin. It

expresses the difference in coulomb interaction energy of the

systems when the electron spins are parallel, or antiparallel.

Because of the Pauli exclusion principle we can not change the

relative direction of two spins without changing the spatial

distribution of charge. If two spins are parallel, the spatial

part of the wave function must be symmetric under the exchange

* W. Heisenberg, Z. Physik 49, 619(1928)

-3

of the two electrons. If the spins are antiparallel, the

spatial part of the wave function is antisymmetric. The

resulting changes in the coulomb energy of the system may be

written in the form 1.11, as if there were a direct coupling

between the directions of the spins Si, Sj.

In contrast to the dipolar coupling between spins (which

is magnetic in origin, and which is anisotropic) the exchange

interaction is of very short range ( since it arises from the

overlap of electron orbitals), and is isotropic.

We can relate the exchange integral to the molecular

coefficient a , and the Curie temperature Tc, by the following

equations(1):

5= g2rE2

5= 3K Tc /2 x J(J+1)

considering the atom under consideration has Z nearest

neighbours, each connected with the central atom by the

interactions. For more distant nei .,hi ours we have taken I as

0. Exchange coupling arises from the overlap of the wave-

functions of different electrons. However calculations of

the overlap of wave-functions on neighbouring atom in the

rare earth metals show that it is too small to account for

the very strong exchange observed. We must conclude that

electrons other than localized 4f electrons take part in the

interaction although they may not contribute to the total

moment per atom. These are the conduction electrons which

act as coupling between the magnetic ions. In fact the magnetic

ion is thought to polarize the conduction electrons around it

44.

and this polarization then acts on other magnetic ions. We

shall discuss this indirect exchange interaction in chapter 2.

1.3.2 SPIN WAVES

Consider a ferromagnetic specimen at absolute zero.

Assume that an axis of quantization is established, say by

a small magnetic field applied along the negative Z direction

The third law of ther4ynamics requires that the spin system

be completely ordered. Since the system must also be in its

ground state, it follows that the spin quantom number of each

atom will have its maximum value. Next suppose that the

temperature is raised slightly so that one spin is reversed;

this presumably is the lowest excited state of the system.

Now , each atom has an equal probability of being the one

whose sain is reversed. This suggest that the reversed spin

will not remain localized at one atom. However, for the

moment consider that the reversed spin is located at a parti-

cular atom. The exchange force will tend to invert the reversed

spin . One possibility is a transition back to the ground

state; this, however, is relatively unlikely . Instead, it

turns out that the reversed spin travels from one atom to

another, the exchange always occuring between neighbours.

The elementary excitations of a spin system have a wave like

form and are called spin waves(13)or, when quantized, magnons.

These are analogous to lattice vibrations or phonons. Because

of the boundary conditions only certain wave lengths are

possible.

45.

Now suppose that as a result of a further increase in

temperature the crystal has two reversed spins. Two

additional complications occure. First, because in general

the two reversed spins, or spin waves, will be travelling

with different velocities, they will meet at some time.

The result is a scattering. Second, there is possibility that

the reversed spins will be bound together on adjacent atoms.

This state, sometimes called a spin complex(14), has a lower

exchange energy than when the two reversed spins are seperated.

If more than two spins are reversed, the same types of

complication occur, although now there will be more collis ions

and also the spin complex may consist of more than two reversed

spins. The usual approxim ation in spin-wave theory is to

neglect these complications and to assume that the spin waves

are independent of each other. This superposition can be

expected to be valid only as long as the number of reversed

spins is small, that is, for temperatures well below the

Curie temperature. If we assume that the Hamiltonian consists

of only the exchange term given by equation 1.11, then

applying the normal treatment to elementary excitations,

we get the following relations for the change in the spon-

taneous magnetisation because of the excitation of spin

waves15-18)

,a M = M(0)-M(T) = 0.1174 ( KT

M(o) r1(o) f (1.14)

Where f = 1,2, and 4 for the simple, body centered, _and face-

centered cubic lattice, respectively, and j is the exchange

46.

energy. This equation is known as the Bloch' T3"2 law.

(1.3.3) BAND MODEL Or FERROMAGNETISM

The preceding theories of ferromagnetism have all been

based on the Heisenberg model in which it is assumed that

the electrons are localized at the atoms. Since the ferro-

magnetic materials are either metals or alloys, it is

obvious that this assumption is invalid-Theories that consider

mobile electrons or holes in unfilled bands have been developed.

calculations in which the interactions between the

electrons of an electron gas are considered have become known

as collective electron theories, the earliest theory considered

the free electron gas(19). It was shown that because of -

correlation effects it was very unlikely that ferromagnetism

would result. Subsequent theories consider the interaction

between electrons and ion cores: that is, they employ Bloch-

type wave functions. The first calculations based on the band

model were made by Slater(20), he obtained result for nickel

that were in fair agreement with experiment. Also Stoner(21)

has initiated a theory known as collective electron ferro-

magnetism. One of the main achievements of the theory of

collective electron ferromagnetism is the prediction'of non-

integral, consitent values for neff, the effective'._number of

magnetic carriers, mentioned in section '1.3 (see table1.2).

In some cases this theory is in better agreement with

experiment than the simple molecular field treatment described

in section 1.3. Detailed comparison with experimental

47.

measurements of magnetic and thermal properties and with

neutron diffraction studeis show that for nickel the collective

electron theory of magnetism is favored, whereas for Gd or

Fe the Heisenberg theory is better. Friedel(22)has proposed

a model that is intermediate between Heisenberg localized

model and Stoner's band model.

1.3.4 CRYSTALLINE ANISOTROPY

The Heisenberg exchange energy depends on the scalar

product Si, Sj, which is invariant with respect to the choice

of coordiAate system. Thus, until now, the magnetisation of

a ferromagnetic specimen has been considered isotropic .

Experimentally, however, it is found that the magnetisation

tends to lie along certain crystallographic axes; this effect

is known as magneto-crystalline anistropy. It is easier to

magnetise a ferromagnet along certain crystalographic axis

(the easy axis) than other axis.

One source of anisotropy is the dipole-dipole interaction.

However in most materials this is not the major source of

anisotropy. An important source of anisotropy is the spin-

orbit interaction. The spin of an ion is coupled to the orbital

motion by spin-orbit coupling, and the orbital motion is

sensitive to the crystalline electric field (the direction

of which is determined by the crystal structure). This source

of anisotropy can be divided into two sub-classes, the first

restricted to cases involving the interaction of the crystal

field with a single ion ( "single-ion anisotropy`! ),and the

48.

second involving anisotropies associated with two or more

spins ("anisotropic exchange"). In the usual(isotropic)

exchange, the spin-orbit interaction is neglected.

Crystalline anisotropy energy, sometimes called magneto-

crystalline energy, is defined as the work required to make

the magnetisation lie along a certain direction compared to

an easy direction. If the work is performed at constant

temperature, the crystalline anisotropy energy is actually

a free energy, to be minimized together with the other

contributions to the total energy of the system. ( For a

complete discussion of anisotropy see for example(23)and

references there in.)

1.4 ANTIFERROMAGNETISM

An antiferromagnet materials has been defined as one in

which antiparallel arrangement of the strongly coupled atomic

dipoles is favored. Neel(24) originaly envisaged an antiferro-

magnetic substance as composed of two sublattices, the spins

of one tending to be antiparallel to those of the other. He

assumed the magnetic moment of the two sublattices to be equal

so that the net moment of the materials was zero. Since N el's

original hypothesis the term antiferromagnetism has been

extended to include materials with more than two sublattices

and those with triangular, Spiral, or canted spin arrangements;

the latter may have a small nonzero magnetic moment. The most

direct method of probing these various spin arrangemnts is

neutron diffraction.

49.

1.4.1 THE MOLECULAR FTRLD MODEL OF ANTIFERROMAGNETISM

The molecular field theory for the simplest case, namely

an antiferromagnetic material with two sublattices was developed

by Neel. (25) Lidiard(26) has calculated the susceptibility of

a single crystal anti'iFromagnet specimen when the molecular

field constant is zero for the similar sites. His results are

shown in Figj,7.,where, X~ is the susceptibility of the speci-men for an applied magnetic field parallel to the easy axis,

and Xis the susceptibility of the specimen for an applied

magnetic field prependicular to the easy axis. TN is the Neel

temperature above which there is no spontaneous magnetisation

and the material is a paramagnet.

If the material is polycrystalline, it is reasonable to

assume that the easy direction in the specimen are randomly

distributed. The susceptibility then can be given by :

xp = x11 +3xl 1.15 In comparison of experimental results with theory the ratio of

the)( at absolute zero to)( at the Neel temperature,/~tp(0)

/gy p p

\I-(T-a) is often considered. The molecular field then predicts X

that :

Xp(o)

Xp(TN)

1,0

x(T)

X (TN)

50.

0.5 to '.5 7/TN

Fig. 1.7. The susceptibility of an antiferromagnetic materials

as a function of temperature in reduced units. ( After ref.26).

51.

1.5 THE DEMAGNETIZATION FACTOR D

The field H' inside a specimen is different from the

applied field H because of the magnetization or eauiv1ently

the poles. Consider a ferromagnet with ellipsoidal shape in

a uniform external field H. The magnetization of the ellipsoidal

specimen will also be uniform. The poles appear on the surface

indicated in Fig. 1.8 produce a uniform internal field H'

opposite in direction to H. For specimens with an ellipsoidal

shape it is usual to write

H' = H - DM,

where D is called the demagnetization factor. D depends on the

geometry of the specimen. For diamagnets H'? H; for all other

magnets H' < H. The difference in the field H' and H can usually

be neglected for dia- and paramagnets, but it can be very large

for ferro-and ferri- magnets. For a disk D = 4 for the direc-

tion prependicular to the plane of the disk. In general the

demagnetizing factor is a tensor.

Because of their practical usefulness, some of the impor-

tant formulas for the demagnetization factor of ellipsoids of

revolution are given.

If we define a as the polar semiaxis and b as the equato-

rial semiaxis with m = a/b. Then for the prolate spheroid(m2> 1)

47C D 1n [in. +(m2 -1)1 -1 Da

(m2 1) (m2-1)

and Db = .2(47C- Da),

where Da is the demagnetization factor for a and Db along b.

For the sphere D =

52.

" REFERENCES "

(1) . Morrish," The Physical Principles of Magnetism " (2) . Brailsford, " The Phisical Principles of Magnetism " (3) . Leighton, " Principles of Modern Physics " (4) . J. H. Van Vleck, " Theory of Electric and Magnetic

Susceptibilities." (5) . K. N. R. Taylor and M. I. Darby, " Physics of Rare Earth

Solids " (6) . W. Pauli, - Z, Physik, 81 (1927) (7) . C. Kittel, " Introduction to Solid State Physics " (8) . L. Brillouin, J. Phys. Radium. 8, 74 (1927) (9) . H. R. Woltjer and K. Kamerleingh Onnes, Commun. Kamerleingh

Onnes Lab. Univ. Leiden, 167C (1923) (10). W. E. Henry, Phys. Rev. 88, 559 (1952) (11) . P. Weiss, J. Phys. 6, 667 (1907) (12). D. H. Martin, " Magnetism in Solids " (13) . F. Bloch. Physik 61, 206 (1930) (14). H. A.•Bethe and A. Sommerfold, Handbuck der Physik,

XX IV/2, J. Springer, Berlin,(1933) P 333. (15). G. Heller and H. A. Kramer, Proc. Roy. Acad. Sci.(Amdterdam

37, 378 (1934) (16) . C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951) (17) : F. Keffer, H. Kaplan,, and Y. Yaft, Am. J. Phys. 21 ,250(1953 (18). J. Van Kranendank and J. H. Van Vleck, Revs. Mod. Phys.

30, 1 (1958) (19) . L. Brillouin, J. Phys. Radium 3, 565 (1932) E.P. Wigner,

Phys. Rev. 46, 1002 (1934), Trans. Faraday Soc. 34. 678

(1938) - (20). J. C. Slater, Phys. Rev; 49, 537 (1936), 49, 931(1936),

52, 198(1937) , Rev. Mod. Phys. 25, 199(1953) (21). E. C. Stoner, Proc. Rev. Soc. (London) A-165, 372(1938),

A-169, 339(1939), Phil. Mag. 25, 899(1938) (22). J. Friedel, G. Leman, and S. Olszenski, J. App. Phys.

32, 3255 (1961)

53. (23) . C. Kittel, " An Introduction to Solid State Physics " (24) . L. Neel, Ann. Phys. (Paris) 17, 64- (1932) (25) . L. Neel, Ann. Phys. (Paris) 18, 5 (1932) 5, 232 (1936);

F. Bitter, Phys. Rev. 54, 79 (1938); J. H. Van Vleck, J. Chem. Phys. 9,85 (1941)

(26) . A. B. I,idiard, Rept. Prog. Phys. 25, 441 (1962)

CHAPTER 2

RARE EARTH METATIS

2.1 Structure behaviour of rare earth metals and alloys

2.2 Magnetic properties

a) Spin contribution

2.2.1 The indirect exchange interaction or R.K.K.Y interaction

b) Orbital contribution

2.2.2.1 The crystal field magnetism

2.2.2.2 Magnetostriction and elastic energy

2.2.3 Magnetic ordering

Thermal first order transition from spiral to:ferro-

magnetic arrangement

2.3 Transport properties (electrical resistivity)

2.3.1 Spin disorder resistivity

2.3.2 Spin wave scattering

2.3.3 The effect of superzone boundaries

2.3.4 Crystal field effects

2.3.5 The effect of alloying

a) Dilute alloys

2.3.5.1 Kondo effects

2.3.5.2 Crystal field effect

b) More concentrated alloys

2.3.5.3 Spin glasses References

5'

The rare earth metals are very similar chemically,

and in many of their physical nrooerties, but have very

different magnetic properties. As is well known, the reason

is that the major part of their chemical and physical

behaviour is determined by the 5d and 6s valence electrons,

while the successive filling of the 4f shell in the rare

earth series is responsible for the rich variety of their

magnetic properties . Perhaps the most striking manifes-

tation of this variety is the Qualitative difference

between the magnetic behaviour of the light and heavy rare

earth metals, but it may also be observed in substantial

differences between the magnetism of neighbouring elements,

which are otherwise very similar. The fact that the rare

earths display the largest. magnetic moments, magnetic

anisotropies and magnetoelastic effects which are known,

makes it possible, by alloying them together, to produce

substances with a wide range of magnetic properties.

2.1 STRUCTTR.nL BEHAVIOUR OF RARE EARTH METALS AND ALLOYS

The structure of all the rare earths at normal

temperatures, with the exception of europium, are of a close

racked nature, and may be described in terms of stacking

sequences involving three types of layers. These may he

defined as A, B, and C and are shown in Fig. 2.1.

In the rare earth all the elements but ytterbium, which

is not a typical member of the series, have room temperature

structure which is the h.c.p. type. This structure has a

55.

stacking sequence A B A B.

At the room temperature the light rare earths mostly

have, d.h.c.p. structure. Lanthanum, Praseodymium and

Neodymium have the double hexagonal structure at room

temperature, having layer stackings ABACA this

corresponds to a stacking fault appearing in every fourth

layer and leads to a doubling of the unit cell C-axis

Parameter. Samarium has a rhombohedral structure which is

unique to this element, although various rare earth alloys

and rare earth metals under pressure possess this structure.

This structure can be expressed in terms of non-primitive

hexagonal unit cell whose C-axis is four and a half times

that of the h.c.p. structure, and having a stacking sequence

ABABCBCAC (Fig. 2.1d ).

Lanthanum, Praseodymium, and Neodymium can be stabilized

also in fcc structure with stacking sequences ABCABC shown

in Fig. 2.1a(1).

ALLOYING BEHAVIOUR

The structures of pure rare earth metals show a systematic

variation through the series from lanthanum to lutetium. By

suitably alloying light rare earths with heavy rare earths,

structures are obtained which are intermediate between those

of the component elements. Fig. 2.2 is the phase diagram

of Y-Nd system as studied by Spedding et al(2). Y can be

considered as an (effectively) heavy rare earth, which has

an electronic energy band structure and lattice structure

A

c

8

A

(a)

B

A

B

A

Cb)

FI G. 2,1

",

A

c

A

C

B

c:.

a

A

Et

"

56.

CC)

(d)

A

C

A

B

A

57.

which causes its alloying behaviour to resemble that of Dy(3).

i'lany authors have attempted to explain the structural

behaviour of the rare earth metals and alloys both quanti-

tavely and qualitatively(4).

Recently Duthie and Pettifor(5) quantitatively correlated

the rare earth crystal structure sequence to the d-band occu-

pancy through the d-band energy contribution to the total

energy. Fig 2.3 shows their result for relative band energies

of h.c.p., d.h.c.p. and Sm structures with ideal axial ratio

with respect to the f.c.c. structure as a function of the

d-band occupancy. We see that, as the d-band progressively

filled with electrons, we move throughout the sequence h.c.p.

--/Sm type - . d. h. c . p ..-.---,f . c . c . as is experimentally observed.

2.2. MAGNETIC PROPERT-17,8

We have seen in previous chapter how Hund's rules deter-

mine the magnetic properties of rare earth ion with an incomple-

tely filled_ 4f shell,the great variety of magnetic structures

for the rare earth metals can be understood as the consequence

of two types of interaction for the l0.calized rare earth ion

moments

H.=Hiso-exc+Horb (2.1)

The first contribution in 2.1, arises from a long range

oscillatory, exchange interaction of the Ruderman-Kittel(6)

type. Via polarization of the conduction electrons._So long_

as one does not explicitly take into account the way in which.

the presence of an orbital contribution to the ionic._ foment

modifies this.interaction(i.e. in practice the limit that the

product of the effective radi' us of 4f orbital wave function

and Fermi radius for the conduction electrons ( in k-space)

,

--

i ,- .

,.- bc.c I I

Ī 1 1 1 I , r 1 I

1 I

h.c•p

d•h•c•p 1' r fi, - 4 II' I I 1

1 1

)

I I l I I 1 I , 1

I I I

1 1 1 1

I I I 1 1ST( I II I I

1 1

) 11 II

IJ 1 20 40 6

atomic 0/0 Y FIG. 2.2

1500

1

50

2

Nd

58.

1 00 Y

Nd

•01

—41

C6:1.58

Fig 2.3. The relative bonding energies of h.c.p( ),

d.h.c.p (---) and Sm type (-----) with respect to the

f.c.c structure as a function of d-band occupancy Vd(5).

59.

is negligible), the Ruderman-Kittel interaction depends only ion

on the scalar product of the total spins of the two interacting

Riso-ex= - Z J (Ri-R~)Si. Si s

The second contribution in 2.1 consist of those interac-

tions whose presence depends on the orbital contribution to

the ionic moment. These interactions are characteristically

anisotropic with respect to the crystal axes and /or depend

on the elastic strains.

Rorb = Ran-ex + H Qf + gm. s (2.2)

The first term is the anisotropic exchange(718)resulting

from taking account of the nonsphericity of a 4f wave function

of finite radi us. The detailed theory of such interaction is

quite complex and depends greatly on the particular exchange

mechanism,Viz, direct via polarization of conduction electrons,

superexchange.

The second contribution in 2.2 is the anisotropy energy

of the unstrained lattice resulting from interaction with the

crystalline electric field caused by each rare-earth ion seeing

the other charged rare earth ions. The crystal field exhibits

the symmetry of the ionic lattice. For the h.c.p. lattice

pertinent to the heavy rare earths, the crystal field interac-

tion consists of a large axial and smaller planar anisotropy.

Rc.f=~ d2 Y2(Ji)+V~ Y4(Ji)+Vō Yō(Ji)+V6 f Y (J ~Y 6 (J3 ),}

(2.2.a) vi

The Ym(J.) are operator equ n ants of spherical harmonics as

discussed by Elliott(9).

60.

The final contribution to- orb comes from magneto-

striction effects. There are both single - ion and two -

ion contributions to the magnetostriction effects which

arise from modulation by the strain of the crystal-field

and anisotropic exchange interactions, respectively.

Hm. s = He + Hm (2.2.b)

Here He is the elastic energy associated with the homogenous

strain components, and Hm is the magnetoelastic interaction,

coupling the spin system to the strains. In the following

subsections we will explain each contribution to the

Hamiltonian in more detail and finally we will see how the

Hamiltonian of 2.1 can lead to various types of magnetic

structures found in the rare earth metals.

2.2.1 THE INDIRECT EXCHANGEINTERACTION OR R.K.K.Y(6) INTERACTION

The exchange interaction between the 4f spin S localized

at a site R and a conduction electron of spin sat position

r is called the S-f exchange interaction and is given by the

familiar Heisenberg form :

Hs-f = - A(r-R) S.s (2.3)

Where A(r-R) is the exchange integral. This interaction is

a straight forward consequence of the pauli exclusion

61.

principle (chapter 1 section 1.3.1). This interaction causes

the polarization of the conduction electron gas. The polari-

zation produced by one ionic spin at Ri will interact with

another spin at Rj through Hs_f . The net result is an

indirect exchange interaction between the localized spin

which, in a first approximiation, also have the Heisenberg

form.

Hip _ - J(Ri - Rfi) Si . S~ (2.L!-)

Where the exchange integral J is a function of the vector

distance Rid = Ri - Rj between the ions. The fourier

transform of this exchange integral J(q) is given by :

iq•(Ri Rfi)

J(q) = J(Ri R.j)e ij

J(q) in terms of s-f exchange integral and the properties

of conduction electron is given by :

J(q) .A2(q)X(q)

Where A(q) is the fourier transform of the s-f exchange

integral, A(r - $) in equation 2.3, and X(q) is the fourier

transform of the non-local susceptibility of the conduction

electron gas and can be calculated from the electronic band

structure. In the R.K.K.Y theory, which has been widely

employed, A(q) is taken to be a constant, Ao say.

62.

The exchange interaction is always between the spins S.

On the other hand the state of a rare earth ion is specified

by its total angular momentum J and it is then necessary to

project S onto J (chapter 1 section 1.2) where the projection

is (g - 1) J(1o). The factor (gj-1) is negative for the

light series and positive for the heavy series ; in consequence

it should be noted that the interaction between a light and

a heavy ion has the opposite sign to the J between them .

Now the Hamiltonian in equation 2.4. can be written as :

Hij = - (g-1)2J(Ri - Rj) Ji.Jj

The above exchange energy depends on the De-Gennes factor

which is defined as (gj-1)2J(J+1). The De-Gennes factor is

generally greater in the heavy rare earth metals than in

the light rare earth metals, indicati ng that the indirect

exchange is more important and stronger in heavy rare earths

than light rare earths.

b) ORBITAL CONTRIBUTION

2.2.2.1 THE CRYSTAL - FIELD NAGivETISH

In this section we will focus our attention on the part

played by crystal electric-field effects in rare earth

magnetism and we will give an example . Crystal field play

a very important role in determining the detailed nature of

magnetic ordering in the heavy rare earth-metals; however

63.

this role is secondary with regard to the question of

w-:e•ther the metals order magnetically at all. For the light

rare earths, crystal - field effects are larger relative to

exchange, and this is no longer true, especially for Pr.

For a rare earth material at temperatures of the order

of the crystal-field splitting, typically tens of K, the

crystal-field can diminish or destroy the orbital contribution

to the ionic moment. However, because the spin-orbit coupling

is very strong, if the crystal-field succeeds in restraining

the orbital moment, the orbital moment also holds back the

spin moment from following the effect of an applied magnetic

field; and thus the crystal-field diminishes or destroy the

total ionic moment.

In this section we will discuss the situation when crystal

field effects, which tend to destroy the ionic moment and hence

tend to destroy magnetic ordering, are comparable to, or

dominant over, the exchange effects which tend to have exactly

the opposite effect. Such a situation can lead to a number

of interesting magnetic properties. The most striking

situation occures when the crystal field only ground state

of the rare earth ion is a singlet. (This can occur only for

non-framers ions, i.e. those rare earth ions with an even

number of f electrons and thus an integral J for the ground

state multiplet.) Then as exchange increases, magnetic

ordering at zero temperature occurs not through the usual

process of alignment of permanent moments, but rather through

a polarization instability of the crystal field only singlet

ground state wave function.

64.

For such induced moment systems, there is a threshold

value for the ratio of exchange to crystal-field interaction

necessary for magnetic ordering even at zero temperature(1117)

we shall discuss the behaviour as this threshold is approached

and exceeded.

We will begin this section with a discussion of the

theoretically expected behaviour for singlet ground state

systems. The way in which magnetic ordering occurs will be

demonstrated with molecular field theory.

ThEORY OF SIl GLET GROUND ST .TE EkGNETISN

Fig. 2.4 shows the crystal-field splitting of the ground

state multiplet Pr3+placed in sites of hexagonal or cubic

symmetry. The crystal-field levels are labeled with their

degeneracies and symmetry types. The ion have integral value

of J, the total angular momentum, in its ground state

multiplets; and the figure shows the common case where the

crystal-field ground state is a singlet. In a hexagonal

crystal-field the first excited state is also a singlet,

while in a cubic crystal-field the first excited state is

a triplet. Typically the splitting from the ground state to

the first excited state is between 10 and 100K; while the

overall splitting is several hundred K (it is useful to

remember that 1meV = 11.606 K).

The singlet ground state means that the ionic moment

vanishes. In the absence of any exchange effects, we get a

Van Vleck susceptibility at low temperature because,an

( 3) r!3 (g) 15 (3)

65.

applied field admixes some excited state wave function into

the ground state thereby inducing a magnetic moment.

We can ask what happens to magnetisation at low temper-

ture as exchange increases. In perticular, how does magnetic

ordering volimpoing first come about as exchange increases from

zero for such a singlet ground state system's?

We begin our discussion by showing that one can arrive

at many of the gross features of induced magnetic behaviour

with a rather simple molecular field theory.

13

(3) 1" _(3)

(1) r

66.

MOLECULAR FIELD THEORY

We can qualitatively understand how magnetic ordering

occurs in system with crystal-field schemes such as those

shown in Fig 2.4 on the basis of a simple molecular field

as originally treated by Trammel~l1'12)and by Bleany(13).

For simplicity, rather than treating the full crystal-field

level scheme, we follow Bleany(13)and consider the two level

modelshown below :

Where the excited state is also a singlet. Besides

simplicity, this two singlet state model has the advantage

of showing purely induced moment effects. (physically, the

two singlet state model would apply, say for Pr3+in a hexa-

gonal envirement when the other excited states were at much

higher energies.)

We consider the Hamiltonian,

H Vci El(Ri-Ri)Ji•Jj HELL, t 11

This Hamiltonian has a crystal-field term giving for each

ion the states ~o and 11c> with energy splitting , an

exchange term, and a Zeeiimanterm•The exchange is taken to

be isotropic. Actually in real systems with large orbital

contribution to the moment, there may be substantial higher

degree and anisotropic exchange interactions the way in which

67.

these have to taken intth account depends on their particular

form. The isotropic exchange taken here will illustrate the

physical effects of interest.

In the absence of exchange the applied field admixes

the two wave functions, and theuby induces the Van Vleck

susceptibility. By conventional first-order perturbation

theory, the susceptibility per ion is :

_ ( A/2612 ) [1 /tanb(A/2KT)1 cF S L j

(2.5)

a=

( The Z-axis has been chosen so that only the 2- component

of angular momentum JZ , has non zero-matrix element between -

the crystal-field only singlet states. The off diagonal matrix

elements of Jx and Jy vanish. Of course, all diagonal matrix

elements of angular momentum vanish for a singlet state. Such

a choice of axes is always possible for an even-electron

two-singlet-level system(18) a hexagonal system, the Z-axis

is the axis of hexagonal symmetry.

Now in a similar way a molecular field induces a

magnetic moment in the singlet ground state (and the opposite

moment in the excited state.). So if one includes exchange in

the molecular field approximation the magnetisation ( per ion )

is .

2 '(0) N =.3A3- =X cF( H + J

or :

~( Xc.F 244

68.

and defining

A _ 4 S(0)a2

the only change in /̀ 'from 2.5 is the replacement

I~tanh (A /2 KT) '/tanb(A/KT) )-A

Thus the effect of exchange on 1/ in the molecular field

theory(13) as illustrated in Fig. 2.5a, is simply to shift

the curve of /X versus T rigidly downward ( ferromagnetic

exchange ) or upward (antiferromagnetic exchange). The

quantity A giving the shift is just the ratio of exchange

interaction to crystal-field splitting. The value of A for

which the susceptibility diverges at T = 0 gives the threshold

value of exchange for ferromagnetic ordering to occur at T=0

For A exceeding the threshold value, it is possible to

find the magnetisation self-consistently as a function of

temperature. The molecular field Hamiltonian is :

H `EVc► — 2š(0).1 EJ;Z

and the molecular field eigenstates are :

1 off = cos el oc) + s i ne.I9 1 = +coseIic%

A

69.

The energy eigenvalues are :

E =-E=-4(Cos2e+ AJSin2e) 0 1 2 (Z

and the rotation angle which diagonalizes H_ is given by :

tan 2e = A( J/a )

then for A exceeding the threshold value, we are able to

find the magnetisation self - consistently as a function of

temperature. This is given by :

.1/c{ =51n20 Tanh (A/T)[1/2Cos20+1/2A(J/a)Sin 21:1

This expression reflects the fact that in the absence of

an applied field, the existance of a molecular field presup-

poses the existance of an ordered moment. This leads to a

threshold value of A at T = 0 for a finite J (i.e. ferro-

magnetism) to exist. This value, AF = 1, is the critical

value for the polarization instability of the ground state

wave function giving ferromagnetism and corresponds to the

divergence of the susceptibility.

One can go through the corresponding theory for an anti-

ferromgnet,(chapter one), where the sublattice magnetisation

takes the place of the magntisation. For antiferromagnetism

the threshold value of A depends on the par ticular type of

antiferromagnetic ordering. The simplest situation is that

for which the moment of any ion is antiparallel to the

70.

moments of all neighbors with-which it has exchange interaction

(i.e. most simply when there is exchange with only one type

of neighbor; that exchange is antiferromagnetic; and the

moment alignment is antiparallel to that of all such neighbors).

In that case the- critical value for ant