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Physics of Magnetism

Physical Principles of Magnetism

In order to understand how rocks, or, more correctly, the

magnetic minerals present within the rock, retain a record

of the Earths magnetic field we need to have some

understanding of the basic physics behind the process.

This is inextricable from quantum mechanics. We cannot,

however, cover the complete quantum mathematical

formulations: instead we will try to gain a qualitative /

semi-quantitative understanding of the physical principles

underlying the phenomenon.


The movement of an electrically charged particle produces

a magnetic field. The result is that all materials display

some magnetic properties. The fundamental units of

magnetic charge are dipoles; a combination of positive

and negative charge (m and -m respectively), which

exhibit a dipole moment. Two kinds of electron motion

define the magnetic properties exhibited by an element.

Radius

Current e-

m


Firstly, electrons orbiting the nucleus have an orbital

angular momentum L, such that:

L = Mass x Radius of orbit x Velocity

The orbiting electron forms a loop of current, which

generates a magnetic field (a magnetic moment ):

= iA, where i is the current and A is the area of the loop

This magnetic moment is quantised in units of B (The

Bohr Magnetron):

= m B, where: B = eh/4me (~9.27x10-24 Am2)

(e is the electron charge, me is the mass of the electron

and h is Plancks constant).


Secondly, the electron has a spin and a spin angular

momentum S.

Each electron spins around an axis, and that axis can have

one of two possible orientations: either parallel or

antiparallel to an external magnetic field. This spinning

charge gives rise to a magnetic field:

S = B

The superposition of these forms of electron motion gives

the total angular momentum of the atom. Magnetism in

solids, however, is dominated by the magnetic moment

associated with electron spin.


The magnetic characteristics displayed by an atom depend

on the arrangement of its electrons. Electrons are arranged

around the nucleus of an atom in shells (states of

successively higher energy). Within a shell, electrons exist

in orbitals, which are described by quantum numbers. Each

orbital contains no more than two electrons (Pauli exclusion

principle) and these have opposite spins.

Physics of Magnetism: Electron orbitals

1s2

2s2

3s2

4s2

5s2

6s2

7s2

2p6

3p6

4p6

5p6

6p6

7p6

3d10

4d10

5d10

6d10

4f14

5f14

8s2

Orbitals are filled with electrons in the order of increasing energy.


Electrons occupy specific energy levels, or orbitals, as they orbit the nucleus of an atom.


n the principal quantum number defines the energy level

of the orbital. Electrons with the same n are said to be in

the same shell. Increasing n indicates shells farther away

from the nucleus.

l the orbital quantum number defines the total angular

momentum of the orbital. l can vary from 0 to n-1.

Electrons with l values of 0, 1, 2, 3 are known as s, p, d and

f electrons.

m the momentum quantum number defines the

component of angular momentum in the direction of the

applied field. m is an integer value 1, 0 -1.

S the spin quantum number defines the spin of the

electron. This can be +½ or -½, and is summed over the full

number of electrons in a shell. A full shell has S = 0,

whereas for Fe S=2.


Atomic

Number

Element K L M N

1s 2s 2p 3s 3p 4s

11 Na ↑↓ ↑↓ 6 ↑

12 Mg ↑↓ ↑↓ 6 ↑↓

13 Al ↑↓ ↑↓ 6 ↑↓ ↑

14 Si ↑↓ ↑↓ 6 ↑↓ ↑ ↑

15 P ↑↓ ↑↓ 6 ↑↓ ↑ ↑ ↑

16 S ↑↓ ↑↓ 6 ↑↓ ↑↓ ↑ ↑

17 Cl ↑↓ ↑↓ 6 ↑↓ ↑↓ ↑↓ ↑

18 Ar ↑↓ ↑↓ 6 ↑↓ ↑↓ ↑↓ ↑↓

19 K ↑↓ ↑↓ 6 ↑↓ ↑↓ ↑↓ ↑↓ ↑

20 Ca ↑↓ ↑↓ 6 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓

ES2: Electron orbitals

1s2

2s2

3s2

4s2

5s2

6s2

7s2

2p6

3p6

4p6

5p6

6p6

7p6

3d10

4d10

5d10

6d10

4f14

5f14

8s2


Atomic

Number

Element Inner Shells

3d 4s

21 Sc

1s22s22p63s23p6

↑ ↑↓

22 Ti ↑ ↑ ↑↓

23 V ↑ ↑ ↑ ↑↓

24 Cr ↑ ↑ ↑ ↑ ↑ ↑

25 Mn ↑ ↑ ↑ ↑ ↑ ↑↓

26 Fe ↑↓ ↑ ↑ ↑ ↑ ↑↓

27 Co ↑↓ ↑↓ ↑ ↑ ↑ ↑↓

28 Ni ↑↓ ↑↓ ↑↓ ↑ ↑ ↑↓

29 Cu ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑↓

30 Zn ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓


The net magnetic moment of an atom:

(J) = orbital angular momentum spin angular momentum.

These combine to minimise J and, hence, the net magnetic

moment.

When J = 0, the atom is non-magnetic

When J 0, the atom acts as a magnet.


As the magnetic moment in solids is dominated by the

magnetic moment associated with electron spin, an atom

will possess an overall magnetic moment where there are

unpaired electrons in an orbital (i.e. the spin moments are

not cancelled).

As single electrons are added the spin moments are

combined, and the resultant magnetic moment (from spin)

is at a maximum when the outer shell is half-full, and

decreases as further electrons are added to the outer shell

until it is full.

Physics of Magnetism: Magnetic Susceptibility

Regardless of the arrangements of the electrons a basic

response is common to all materials on the application of a

magnetic field. This is because the applied field exerts an

aligning torque on electron orbits, causing them to rotate

and, hence, producing a magnetisation, which is parallel or

anti-parallel to the applied field. The response is

dependent on its magnetic susceptibility (). Magnetic

susceptibility can therefore be most simply regarded as the

ease with which a material can be magnetised, and is a

dimensionless parameter which links the magnetic moment

of the material with the applied field:

J = H

Where J is the magnetic moment (A/m) and H is the applied

field (Tesla).

Physics of Magnetism: Magnetic Moments

J

H

J=H

Ferro- / Ferrimagnetic Material

Paramagnetic Material

Diamagnetic Material

J=H

Physics of Magnetism: Diamagnetism

Diamagnetic materials are those which, when a magnetic

field is applied, acquire a small induced magnetization

opposite to the applied field (e.g. Quartz). The induced

magnetization is linearly dependant on the applied field and

decays to zero when the field is removed. Diamagnetism is

a property of all matter but the effect is swamped in

substances whose atoms possess atomic magnetic moments.

Physics of Magnetism: Paramagnetism

J

H

J=H




J=H

Physics of Magnetism: Paramagnetism

Paramagnetic substances are those which, when a magnetic

field is applied, acquire an induced magnetization parallel

to the applied field (e.g. Fayalite; an iron-rich Olivine).

Paramagnetic substances contain atoms with atomic

magnetic moments but with no interaction between

adjacent atomic moments (i.e. atoms with unfilled shells).

Again the magnetization is linearly dependant on the

applied field and decays to zero when the field is removed.

The effect is much stronger than the diamagnetic behaviour

by a factor of about 10 to 100.

Physics of Magnetism: Permanent magnetism

J

H

J=H




J=H

Physics of Magnetism: Ferromagnetism

The transition metals and rare earth elements (and their

compounds) behave as through J = S not J = L S. The

orbital moment is said to be quenched. This occurs because

the 3d (or 4f) electrons (which occupy the outermost

orbitals) have highly eccentric orbitals, which extend

proportionately farther form the nucleus, and interact with

surrounding atoms. These 3d (or 4f) electrons experience

an electrostatic ‘crystal field’ with outweighs the

electrostatic L-S coupling. The Spin dominates.

As the 3d states are filled progressively, the electrons are

added with parallel spins until all 5 orbitals are filled, and

all rotate in unison with the applied field.

There is, however, no linkage between the spin directions in

adjacent atoms.


Atomic

Number

Element Inner Shells

3d 4s

21 Sc

1s22s22p63s23p6

↑ ↑↓

22 Ti ↑ ↑ ↑↓

23 V ↑ ↑ ↑ ↑↓

24 Cr ↑ ↑ ↑ ↑ ↑ ↑

25 Mn ↑ ↑ ↑ ↑ ↑ ↑↓

26 Fe ↑↓ ↑ ↑ ↑ ↑ ↑↓

27 Co ↑↓ ↑↓ ↑ ↑ ↑ ↑↓

28 Ni ↑↓ ↑↓ ↑↓ ↑ ↑ ↑↓

29 Cu ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑↓

30 Zn ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓


Ferromagnetic solids have atoms with atomic magnetic

moments which strongly interact with each other (e.g.

magnetite). In these materials the atoms are packed in the

crystal lattice in such a way that the orbitals of adjacent

atoms overlap (in the case of iron (Fe) the 3d orbitals are

highly eccentric and overlap) . This causes the electrons to

try to adhere to the Pauli exclusion principle of both atoms

simultaneously, which effectively means that electrons are

shared between adjacent atoms. This results in strong

parallel coupling of electron spin moments throughout the

material and these aligned moments give rise to a strong

permanent magnetisation.


In pure ferromagnetic materials, covalent bonding occurs by

exchange of one or more 3d electrons shared between

adjacent atoms, forcing them to have parallel 3d orbital

spins. This is only favourable in atoms with more than five

3d electrons as the sharing then brings them closer to a

‘noble gas’ state. E.g. CrO2 which was used extensively in

audio tapes.

↑ ↑ ↑ ↑ ↑ ↑ ↑

Cr – 3d orbital Oxygen

Physics of Magnetism: Anti-ferromagnetism

In most oxides and sulphides of ferromagnetic material, the

oxygen provide a link between nearest neighbour Fe

cations, which force the atomic dipoles of the Fe cations to

be anti-parallel. FeO (wustite is an example)

↑ ↑

Fe2+ O2-

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

Fe2+

Physics of Magnetism: Ferrimagnetism

In most oxides and sulphides of ferromagnetic material, the

oxygen provide a link between nearest neighbour Fe

cations, which force the atomic dipoles of the Fe cations to

be anti-parallel. Fe3O4 (Magnetite is an example)

↑ ↑

Fe3+ O2-

↑ ↑ ↑ ↑ ↑ ↑ ↑

Fe2+


J

H

J=H




J=H


In Ferromagnetic materials applied magnetic fields induce a

magnetization parallel to the applied field, which can be

retained after removal of the applied field, giving rise to a

remanent magnetization. The magnetization does not

exhibit a linear relationship with the applied field and for a

given ferromagnetic material and temperature there is

maximum magnetization, known as the saturation

magnetization (Js), beyond which an increased applied field

will not increase the induced magnetization. The saturation

magnetization decreases with increasing temperature until

it is reduced to zero at a temperature known as the Curie

temperature, which is characteristic of the particular

ferromagnetic material. At temperatures above the Curie

point the material exhibits paramagnetic behaviour.

Physics of Magnetism: Hysteresis

J

H

JS

JR

HC

Hysteresis

Loop

When a ferromagnet is subjected to a

cyclic change in the external field the

magnetisation is not directly

proportional to the applied field by

there is a lag in the magnetisation,

which is known as hysteresis. H is the

applied field, J is the induced

magnetization. Js is the saturation

magnetization, Jr is the saturation

remanence and Hc is the coercivity.

The various hysteresis properties are

not solely intrinsic properties but are

dependent on grain size, domain

state, stresses and temperature.

Because hysteresis parameters are

dependent on grain size, they are

useful for magnetic grain sizing of

natural samples.


In ionic compounds, such as oxides, more complex forms of

magnetic ordering can occur. Such compounds can have two

atomic sublattices. If the ferromagnetic effects within

these sublattices oppose and exactly cancel out each other

the material is antiferromagnetic. Ferrimagnetic and

canted antiferromagnetic substances are those where the

two internal ferromagnetic effects do not completely cancel

each other. These materials behave like ferromagnetics and

have a Curie temperature (or more correctly a Néel

temperature).

Physics of Magnetism: Temperature Effects

Alignment of magnetic moments at various temperatures: at

0°K there is perfect alignment, but above this the spin

moments precess about the average direction due to

thermal activation. Above the Curie temperature they are

random.

0°K T > Tc 0°K < T < Tc


Magnetic Properties

of all materials

Permanent

magnetic moment?

Long-range order?

Nearest-neighbour

orientation?

Magnitude of anti-

parallel moments?

Diamagnetism

Paramagnetism

Ferromagnetism

Ferrimagnetism

Antiferromagnetism

No

No

Equal

Yes

Yes

Antiparallel

Parallel

Unequal

Magnetism in Oxides

Magnetism in Oxides

Titano-Magnetite

series

Ilmeno- Haematite

series

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