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