Transcript Physics of Magnetism - University of Oxford

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.
Physics of Magnetism
m
Radius
e-
Current
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.
Physics of Magnetism
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/4me (~9.27x10-24 Am2)
(e is the electron charge, me is the mass of the electron
and h is Plancks constant).
Physics of Magnetism
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.
Physics of Magnetism
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
2p6
3s2
3p6
3d10
4s2
4p6
4d10
4f14
5s2
5p6
5d10
5f14
6s2
6p6
6d10
7s2
7p6
8s2
Orbitals are filled with
electrons in the order of
increasing energy.
Physics of Magnetism
Electrons occupy specific energy levels, or
orbitals, as they orbit the nucleus of an atom.
Physics of Magnetism
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.
Physics of Magnetism
Atomic
Number
Element
K
1s
L
2s
11
12
13
14
2p
M
3s
3p
Na
Mg
Al
Si
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
6
6
6
6
↑
↑↓
↑↓
↑↓
↑
↑ ↑
15
16
17
P
S
Cl
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
6
6
6
↑↓
↑↓
↑↓
↑ ↑ ↑
↑↓ ↑ ↑
↑↓ ↑↓ ↑
18
19
20
Ar
K
Ca
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
6
6
6
↑↓
↑↓
↑↓
↑↓ ↑↓ ↑↓
↑↓ ↑↓ ↑↓
↑↓ ↑↓ ↑↓
N
4s
↑
↑↓
ES2: Electron orbitals
1s2
2s2
2p6
3s2
3p6
3d10
4s2
4p6
4d10
4f14
5s2
5p6
5d10
5f14
6s2
6p6
6d10
7s2
7p6
8s2
Physics of Magnetism
Atomic
Number
Element
21
22
23
24
Sc
Ti
V
Cr
25
26
27
28
29
30
Inner Shells
3d
4s
↑
↑ ↑
↑ ↑ ↑
↑ ↑ ↑ ↑ ↑
↑↓
↑↓
↑↓
↑
Mn
Fe
Co
↑ ↑ ↑ ↑ ↑
↑↓ ↑ ↑ ↑ ↑
↑↓ ↑↓ ↑ ↑ ↑
↑↓
↑↓
↑↓
Ni
Cu
Zn
↑↓ ↑↓ ↑↓ ↑ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑↓
↑↓
↑↓
↑↓
1s22s22p63s23p6
Physics of Magnetism
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.
Physics of Magnetism
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
Ferro- / Ferrima gnetic M a teria l
Pa ra ma gnetic M a teria l
J= H

H
Dia ma gneti c M a teria l
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
Ferro- / Ferrima gnetic M a teria l
Pa ra ma gnetic M a teria l
J= H

H
Dia ma gneti c M a teria l
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
Ferro- / Ferrima gnetic M a teria l
Pa ra ma gnetic M a teria l
J= H

H
Dia ma gneti c M a teria l
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.
Physics of Magnetism
Atomic
Number
Element
21
22
23
24
Sc
Ti
V
Cr
25
26
27
28
29
30
Inner Shells
3d
4s
↑
↑ ↑
↑ ↑ ↑
↑ ↑ ↑ ↑ ↑
↑↓
↑↓
↑↓
↑
Mn
Fe
Co
↑ ↑ ↑ ↑ ↑
↑↓ ↑ ↑ ↑ ↑
↑↓ ↑↓ ↑ ↑ ↑
↑↓
↑↓
↑↓
Ni
Cu
Zn
↑↓ ↑↓ ↑↓ ↑ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑↓
↑↓
↑↓
↑↓
1s22s22p63s23p6
Physics of Magnetism: Ferromagnetism
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.
Physics of Magnetism: Ferromagnetism
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)
↑ ↑ ↑ ↑
↑ ↑ ↑ ↑ ↑
Fe2+
O2-
Fe3+
Physics of Magnetism: Magnetic Susceptibility
J
Ferro- / Ferrima gnetic M a teria l
Pa ra ma gnetic M a teria l
J= H

H
Dia ma gneti c M a teria l
J= H

Physics of Magnetism: Magnetic Susceptibility
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
JS
JR
HC
H
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.
Physics of Magnetism
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
0°K
0°K < T < Tc
T > Tc
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.
Physics of Magnetism
Magnetic Properties
of all materials
Diamagnetism
No
Permanent
magnetic moment?
Yes
Paramagnetism
No
Long-range order?
Yes
Ferromagnetism
Parallel
Nearest-neighbour
orientation?
Antiparallel
Ferrimagnetism
Antiferromagnetism
Unequal
Magnitude of antiparallel moments?
Equal
Magnetism in Oxides
Magnetism in Oxides
IlmenoHaematite
series
TitanoMagnetite
series