TYPES OF INTERACTION IN MAGNETISM

Ferromagnetism is a type of magnetism characterized by an spontaneous parallel alignment of atomic magnetic moments, with long range order. Examples of ferromagnets are the elements iron, nickel, and cobalt. This order disappears above a certain temperature called the Curie temperature. From the different interactions taking place in atoms constituing a solid, the question is which of those shows responsible for macroscopic magnetic effects? A quick look at the dipolar Interaction will reveil that this type cannot account for long range order. But exchange interaction, looked at in section 3.2, presents a valid explanation: The energydifference between a singlet and a triplet state is at the very core of explaining the long range order. So in the end magnetism on the bigger scale is a purely quantum mechanical effect.

MINISTRY OF EDUCATION AND TRAINING

HUE UNIVERSITY COLLEGE OF EDUCATION

I’m grateful to my supervisor, Dr PHAM HUONG THAO for helping during my study My would not have been complete without her guidance at the beginning and the fruitful discussion later on Her continual encouragement careful reading, critical comments and patient guidance made my work more enjoyable and easier.

I would like to thanks all teachers in Physics department of Hue University’s College of Education and Foreign teachers for teaching and helping me in my courses.

Lastly, I would to acknowledge to my family, my friend and my classmate Their love, support and constant encouragement gave me a great deal of strength and determination that help me during the stressful time of writing this paper.

It is my great pleasure to thanks these people.

Student

Le Thi Hoai

TABLE OF CONTENTS

LIST OF FIGURES

Figure 1: Vector model of the atom: The plane of the electron’s orbit can have

only certain possible orientations, we say it is spatially quantized [Reprinted from

H Lueken, Magnetochemie, 1 Auflage; Teubner Verlag]……….…………12

Figure 2: Magnetic moment due to a current loop [Reprinted from Stephen

Blundell, Magnetism in Condensed Matter, 1

Edition; Oxford Univ Press]……….14

Figure 4: Electron spin in a magnetic field Bz

[Reprinted from H Haken, H.C.Wolf; Atomund Quantenphysik, 8 Auflage; Springer Verlag]………16

Figure 5: Vector model of LS-coupling [Reprinted from Wolfgang Demtröder,

Experimentalphysik Band 3, 3 Auflage; Springer Verlag]………20

Figure 6: Hund’s rule assume combination to form S and L, or imply L-S

(Russell-Saunders) coupling……… …………20

Figure 7: Characteristic magnetic susceptibilities of diamagnetic and paramagnetic

substance……….23

Figure 8: Fe, Co and Nickel are ferromagnetic so that they have a spontaneous

magnetization with no applied field [Reprinted from Stephen Blundell, Magnetism

in Condensed Matter, 1

st

Edition; Oxford Univ Press]……….….24

Figure 9: Dipol-Dipol interaction energy for two colinear dipoles with the same

Figure 12: Antiparallel alignment for small interatomic distances……….34 Figure 13: Parallel alignment for large interatomic distances……… ……… 34 Figure 14: The Bethe-Slater curve……… ……35 Figure 15: The coefficient of indirect (RKKY) exchange versus the interatomic

Figure 17: Magnetization curve of iron [Reprinted from Heiko Lueken,

Magnetochemie, 1 Auflage; Teubner Verlag]……….38

INTRODUCTION

1. Reasons for choosing the topic

Even though magnetic phenomena have been known for centuries butmodern physics was able to put them on a solid basis Earlier attempts, especiallyexplaining magnetism on the macroscopic scale, remained in some mysteriousways Only with the birth of quantum mechanics, the magnetic phenomena could

be understood clearly (via exchange interaction)

Today where the world market for magnetic media and recording equipmentreaches billions dollars per year, the magnetic materials, which are the basis of thepresent technological revolution, remains a very interesting and active field ofphysical research

The interactions take an important role in magnetism, especially exchangeinteraction Studying the interactions helps students fill gaps in basic knowledge ofmagnetism

For the above reasons, I write the independent study: “types of interaction inmagnetism “

2. Aims of study

Studying theoretically some fundamental concepts in magnetism asmagnetic moment of atom and some interactions in magnetism (dipolar interactionand exchange interaction)

3. Contents of study

The independent study focuses on the following problems:

 Magnetic moments of single atoms

 Many electron atoms with unpaired electrons

 Types of interaction in magnetism

4. Body of independent study

Additional to the introduction and the conclusion, the graduate thesis’scontent consists of four chapter:

Chapter 1: Magnetic moments of single atoms

Chapter 2: Many electron atoms with Unpaired electrons

Chapter 3: Types of interaction in magnetism

Chapter 4: Other types of interaction

111Equation Chapter 1 Section 1CHAPTER 1 MAGNETIC

MOMENTS OF SINGLE ATOMS

1.1 Quantum Mechanics

1.1.1 The Schrodinger equation in spherical coordinates

The energy levels of an atom are solutions to the Schrodinger equation:

If the momentum p in the experession for the classical Hamiltonian for a one

particle system is replaced by − ∂ ∂ i h / xi

the corresponding operator is obtained:

where V is the potential in which the electron moves

In three dimensions the Schrodinger equation generalizes to:

where

2

∇

is the Laplacian operator

Using the Laplacian in spherical coodinates, the Schrodinger equationbecomes:

And by making use of the Legendre polynomials m(cos )

l

andexp( ),

r n

e m E

e p

m m m

m m

=+

1.1.2 The Quantum Numbers

The theory of quantum mechanics tells us that in an atom, the electrons arefound in orbitals, and each orbitak has a characteristic energy Orbital means “smallorbit” We are interested in two properties of orbitals – their energies and theirshapes Their energies are important because we normally find atoms in their most

stable states, which we call their ground states, in which electrons are at theirlowest possible energies.

• The Principal Quantum Number n:

The quantum number n is called the principal quantum number We alreadyknow this as shell/orbit The shell “K” has been given the value n=1, the “L” shellhas been given the value n=2… the shell are denoted by letters as shown in thetable below

Number n 1 2 3 4 5 6 7

Shell K L M N O P Q

The principal quantum number serves to determine the size of the orbital, orhow far the electron extends from the nucleus The higher the value of n the furtherfrom the nucleus we can expect to find it As n increases so does the energyrequired as well because the further out from the nucleus we go the more energythe electron must have to stay in orbit Bohr's work took into account only this firstprinciple quantum number His theory worked for hydrogen because hydrogen just

happens to be the one element in which all orbitals having the same value of n also

have the same energy Bohr's theory failed for atoms other than hydrogen,

however, because orbitals with the same value of n can have different energies

when the atom has more than one electron

• The Orbital Angular Momentum Quantum number l :

The secondary quantum number, l, divides the shells up into smaller groups of subshells called orbitals The value of n determines the possible values for l For any given shell the number of subshells can be found by l = n -1 This means that for n = 1, the first shell, there is only l = 1-1 = 0 subshells ie the shell

and subshell are identical When n = 2 there are two sets of subshells; l = 1 and l =

0 The value l =0,1,2, ,(n−1 )

For a particular value of l, the magnitude of the total angular momentum ur Lorbit

of an electron due to its orbital motion is given by:

As with the principal quantum number, letters are used to denote specificorbital quantum numbers:

The principle quantum number describes size and energy, but the secondquantum number describes shape The subshells in any given orbital differ slightly

in energy, with the energy in the subshell increasing with increasing l This means

that within a given shell, the s subshell is lowest in energy, p is the next lowest,followed by d, then f, and so on For example:

Figure 1: Vector model of the atom: The plane of the electron’s orbit can have only

certain possible orientations, we say it is spatially quantized [Reprinted from H Lueken, Magnetochemie, 1 Auflage; Teubner Verlag].

• The Spin quantum number ms

:The fourth and final quantum number is used to indicate the orientation of

the two electrons in each orbital The values for ms

are

12

1.2 Magnetic moments

From the expression for the torque on a current loop, the characteristic of

the current loop are summarized in its magnetic moment µur

, due to current loop isgiven by:

is the oriented area enclosed by that loop as depicted in Fig.2

Figure 2: Magnetic moment due to a current loop [Reprinted from Stephen

Blundell, Magnetism in Condensed Matter, 1

st

Edition; Oxford Univ Press]

The magnetic moment can be considered to be a vector quantity with directionperpendicular to the current loop in the right – hand – rule direction

From classical Electrodynamics we have:dµur=Id Sur,

1.2.1 Magnetic moment due to orbital motion of the electron.

In atomic physics, the Borh magneton µB

is a physical constant and thenatural unit for expressing the magnetic moment of an electron caused by either itsorbital or spin angular momentum

If τ

denotes the time the electron needs to complete one full loop, the Borh

magnetic µB

can be defined, which is the smallest magnetic unit possible How it

is calculated is shown in Fig.3?

Circulating current is I:

2 /

e e I

r v

τ π

= =The Borh magneton:

Edition; Oxford Univ Press]

But to find the magnetic moment if the electron is in an excited state, quantumnumbers are needed

1.2.2 Magnetic moment for an electron in an excited state

The magnitude of the magnetic moment associated with the orbitalmomentum of the electron is:

.2

13113\* MERGEFORMAT (.)The relation between the magnetic moment and the orbital momentum can be

is the so called gyromagnetic ratio

So quantum numbers become a factor in the equations for

l

µ

and

l z

µ

thusdetermining the magnitude of the magnetic moment according to the state of thesystem

1.2.3 Angular momentum (Spin) and associated magnetic moment of the electron

The spin quantum number ms

can take the two values

1.2

s

m = ±

So themagnetic moment along a particular axis corresponding to the spin angularmomentum is:

14114\* MERGEFORMAT (.)Which differs from the value of the orbital magnetic moment by the factor 2

(2.0023 for a free electron) This factor is called the g-factor gS

The total magnitude of the spin angular momentum is:

sr = s s( +1 ,)h

white

1.2

s=

15115\* MERGEFORMAT (.)

In the vector model in Fig.4 it is the length of the depicted arrow

The spin quantum number for theelectron:

s= m = ±The component of angular momentum

along a particular axis: 2

h

(up) or −2h(down)

The magnitude of spin angular

So, the component of the intrinsicmagnetic moment along the z axis is:

[Reprinted from H Haken, H.C.

Wolf; Atomund Quantenphysik, 8.

Auflage; Springer Verlag].

This concludes the review on quantum numbers and how they enable us todetermine the angular momenta and associated magnetic moments according to thestate of the system under consideration, where in this section we only had a look atsingle electron systems.

• the nucleus often has a non-zero spin

→ nuclear spin quantum number I

• very small magnetic moment (between

3

10− and

CHAPTER 2 MANY ELECTRON ATOMS WITH

UNPAIRED ELECTRONS

2.1 L Sur ur−

coupling or Russel Saunders coupling

For multi-electron atoms where the spin- orbit coupling is weak, it can bepresumed that the orbital angular momenta of the individual electrons add to form aresultant orbital angular momentum L Likewise, the individual spin angularmomenta are presumed to couple to produce a resultant spin angular momentum S.Then L and S combine to form the total angular momentum:

Jur ur ur= +S L.

16116\* MERGEFORMAT (.)Scalar products of the angular momentum operators describe theseinteractions.For two electrons labeled i and j the three possible interactiosn are:

,,

ii ii

As a result the spins couple to form the total spin S

ur and the orbital momenta

couple to form the total momentum Lur

This is known as the L Sur ur−

model of the LS-coupling and decribes how urL

and S

ur Couple to form J

ur

Listed below are the possible values the quantum number J

urcan take:

 The total atomic angular momentum is: urJ = J J( +1)h

 The total orbital angular momentum is: urL= L L( +1)h

 The total spin angular momentum is: Sur= S S( +1)h

Associated with J is a magnetic momentum µJ

This moment is not simply

the sum of µS

and µL

but is calculate by the Lande-Formula In the case of

Note that J can take the value J =(L S+ ),(L S+ −1), , L S−

This means tha

depending on the value of J also µJ

differs in magnitude

 Magnitude of the magnetic moment in the case of L Sur ur−

coupling:

The following formula is known as the Lande-Formula It yields the projection

of the magnetic moment µJ

onto the vector urJ

This projection is what can bemeasured the total magnitude of the magnetic moment is not accessible through

J J

+uur

Figure 5: Vector model of LS-coupling [Reprinted from Wolfgang Demtröder,

Experimentalphysik Band 3, 3 Auflage; Springer Verlag]

that result in the smallest energy, and therefore is the

most stable configuration, is that for which the quantity 2S +1

When the first rule is satisfied, there are several possible value of L

(for the

same value 2S +1

); the most stable is the one that makes L

maximum

3 For atoms with less than half filled shells, the level with the lowest value

of J lies lowest in energy

having found S, L and J, this ground state can be summaried using a term

symbol of the form

spin-direction, let’s say spin-up

12

spindown electron

12

i

s 1

2

12

12

12

12

12

use term symbol D

• Shell is more than haff full:

5 4

J = + = ⇒ D

With term symbol

5 4

2.3 Dia- and Paramagnetism

Figure 7: Characteristic magnetic susceptibilities of diamagnetic and

paramagnetic substance.

By applying an external magnetic field Huur

on a substance one can observe

an alignment of the microscopic magnetic moments of that substance either parallel

or anti-parallel to the direction of Huur

This alignment is the cause for the

the mass susceptibility χg

.The mafnetic susceptibility per unit volume is defined as:

,

g

χχρ

=

20120\* MERGEFORMAT (.)

where χg

is the mass susceptibility

Substances with a negative magnetic susceptibility are called diamagnetic

Figure 8: Fe, Co and Nickel are ferromagnetic so that they have a spontaneous

magnetization with no applied field [Reprinted from Stephen Blundell, Magnetism

in Condensed Matter, 1

st

Edition; Oxford Univ Press].

The quantum mechanic operator describing magnetic systems is obtained by

second order perturbation theory It is composed of the undisturbed operator

¶0

H

, aparamagnetic term and a diamagnetic term The paramagnetic term only depends

on electrons beeing present at positions r i

r, so every atom/ion possesses a

diamagnetic component For the paramagnetic term it is possible to vanish as urL

and Sur

can both be zero In that case the atom in question would be a diamagnet

(negative χ

) In figure 8 the mass susceptibility χg

for the first 60 elements in theperiodic table at room temperature are given

0 1

2

Z i