Magnetism

UEEP2024 Solid State Physics
Topic 5 Magnetism

Magnetic properties of solids
• The magnetic moment of a free atom has
three principal sources :
1. The spin with which electrons are endowed
2. The electron orbital angular momentum about
the nucleus
3. The change in the orbital moment induced by an
applied magnetic field

Materials may have intrinsic magnetic dipole moments, or they
may have magnetic dipole moments induced in them by an
applied external magnetic field of induction. In the presence of a
magnetic field of induction, the elementary magnetic dipoles,
whether permanent or induced, will act to set up a field of
induction of their own that will modify the original field. The
magnetic dipole moments are the source of magnetic induction B
MHB oo  
Magnetic field strengthMagnetic
constant
Magnetization

Alternative names for B and H
B
name used by
magnetic flux density electrical engineers
magnetic induction applied mathematicians
electrical engineers
magnetic field physicists
H
name used by
magnetic field intensity electrical engineers
magnetic field strength electrical engineers
auxiliary magnetic field physicists
magnetizing field physicists

Magnetization is defined as magnetic moment per unit volume.
For certain magnetic materials, it is found empirically that the
magnetization M is proportional to magnetic field strength H
HM 
Magnetic susceptibility
 > 0 -- paramagnetism
 < 0 -- diamagnetiism

Atomic Magnets
• In classical picture of an atom, the electrons
describe a circular orbit around the nucleus.
• Each orbit can therefore be thought of as a loop
of electric current.
• From electromagnetic theory, a loop of current
produces a magnetic field, so electrons in an
atom also generate a magnetic field.
• Quantum theory predicts that electrons in an
atom produce a magnetic field.
• The quantum number n, l, ml and ms label the
electrons in an atom.

Atomic Magnets
• The orbital magnetic number ml takes values
between –l and +l.
• Spin magnetic number ms takes values of
1/2.
• A single electron the component of the spin
magnetic moment, µs, in the direction parallel
to the magnetic field is given by
• The quantity is known as the Bohr
magneton.
ee
s
s
m
e
m
em
2




e
B
m
e
2



Atomic Magnets
• The component of orbital magnetic moment is
given by
• When an atom has more than one electron, for a
particular subshell,
Total spin angular momentum
Total orbital angular momentum
• If a subshell is filled, the values of S and L are
both zero.
• If a subshell is partially filled, electrons are
distributed between different ml and ms states.
Bl
l
l m
m
em
 
2

 smS
 lmL

Hund’s rules
• Two simple rules known as Hund’s rules to
determine how theses states are occupied in a
partially filled sub band.
1. The states are occupied so that as many
electrons as possible have their spin aligned
parallel to one another, i.e. so that the values of
S as large as possible.
2. When it is determined how the spins are
assigned, the electrons occupy states such that
the value of L is a maximum.

Total angular momentum J
• The total angular momemtum J is obtained by
combining L and S as follows:
 If the subshell is less than half filled then J = L-S;
If the subshell is more than half filled then J = L +
S;
If the subshell is exactly half filled then L = 0 and
so J = S.
• By applying these rules, the values of S, L and J
can be determined.

Landé splitting factor g
• The maximum component of the magnetic
moment of the atom in the direction parallel
to the magnetic field is
where g is the Landé splitting factor
• Most atom exhibit a magnetic moment.
Jgm Bj 
)1(2
)1()1()1(3



JJ
LLSSJJ
g

Example
Determine the values of S, L and J for Cr3+
which has three electrons in the 3d subshell.
(All lower energy subshells are filled)

Solution
• The 3d subshell corresponds to l = 2, and therefore ml takes
integer values between -2 and +2. i.e. there are five
allowed values of ml. so that the subband has a capacity to
hold 10 electrons.
• In Cr3+, the three 3d electrons can therefore all occupy ms =
+1/2 and therefore S = 3/2.
• The maximum value of L is obtained if two electrons occupy
the state with ml = +2 and one occupies the ml = +1 state.
However, the exclusion principle forbids two electrons with
the same value of ms from occupying the same ml state.
• Consequently the largest allowed value of L corresponds to
having one electron in each of the states m = +2, +1 and 0,
and therefore L = 3.
• Since the subband is less than half filled, the value of J is
J = L – S = 3 – 3/2 = 3/2.

Which materials have magnetic
moment?
• Filled electron subshell in an atom do not
affect the magnetic moment because S = L = 0
so J = 0.
• Inert atoms have a magnetic moment of zero.
• Ionic materials have a magnetic moment of
zero because the electrons are transferred
from one atom to another so that the
resulting ions have only filled subshells.

moment?
• In covalent material the outer subshell is only partially
filled, so these materials have finite magnetic moment.
• However, each covalent bond is formed by a pair of
electrons with opposite spin and with a net orbital
angular momentum of zero. Covalent solids have a net
magnetic moment zero.
• Actually, this is not quite true. The present of a
magnetic field also affect the orbital motion of the
electrons in an atom in such a way that the atom
generate a magnetic field which opposes the external
field. This is referred to as diamagnetism.

moment?
• Filled electron subshells in an atom do not affect
the magnetic momentum of the atom because
they have a net angular momentum of zero.
(S=L=0, so J = 0).
• This means that inert atoms have a magnetic
moment of zero because they have only filled
electron subshells.
• Ionic materials have a magnetic moment of zero
because the electrons are transferred from one
atom type to another so the resulting ions have
only filled subshells.

moment?
• In covalent materials the outer subshell is only
partially filled, and so these materials have a
finite magnetic moment.
• However, each covalent bond is formed by a pair
of electrons with opposite spin and with net
orbital angular momentum of zero, covalent
solids have a net magnetic moment of zero.
• Although most atoms have a non-zero magnetic
moment, it appears that in majority of solids the
effects cancel and the resultant magnetization is
zero.

moment?
• The presence of a magnetic field affects the
orbital motion of the electrons in an atom in
such a way that the atom generates a
magnetic field which opposes the external
field. This is referred to as diamagnetism and
occurs in all type of atom.
• It produces a finite magnetic susceptibility in
all solids, but the effect is weak.

moment?
• It appears that most non-metals display only a small
magnetic susceptibility.
• In simple metal, such as sodium and aluminium, the
valence electrons are assumed to be localized, in other
words they are no longer attached to any particular
atom. As a result the metal ions contains only filled
electron subshells, and so each has a total angular
momentum of zero.
• Although the ions do not contribute to the magnetic
moment, the delocalised electrons do produce a non-
zero magnetic moment. This is known as Pauli
paramagnetism.

moment?
• Free electron paramagnetism is a weak effect.
• Paramagnetism is caused by the magnetic
dipole moments becoming aligned with the
magnetic field, whereas diamagnetism results
from an induced magnetic field which opposes
the applied field.
• Paramagnetism gives rise to a positive
susceptibility, whilst diamagnetism produces a
negative susceptibility.

moment?
• A group of metallic elements known as the
transition metal, and the so called rare earth and
actinide elements have more promising
characteristics.
• These materials have an electronic structure
which is very different to that of the simple
metals.
• These elements give rise to Curie paramagnetism.
• Under certain circumstances they can form
permanent magnets.

Diamagnetism
• The value of B is smaller in the region of the
diamagnetic material than it would be if the
material were absent
• The origin of diamagnetism is Lenz’s law
When the flux through an electrical circuit is changed,
an induced current is set up in such a direction as to
oppose the flux change.

Diamagnetism
(Classical theory)
2
omRF 
Consider an electron in a
circular orbit of radius R and
angular frequency o
If a magnetic field dB is turned
on, and assume it affects
angular frequency only
BqvmRF o d  2

Diamagnetism
(Classical theory)
   
    BqRmRmR
BqRmRmR
BqvmRmR
oooo
ooo
o
dddd
ddd
d



222
22
22
2
The new angular frequency 
Taking only first order form (assume o>>d)
m
Bq
BqRmR oo
2
2
d
ddd 
Classically, the B field will make the electrons revolve slower

Diamagnetism
(Classical theory)

d
d


22
qIqI 
Current formed by the loop = charge x revolution/second
For an electron in an atom
m
Be
I
m
Bqq
I

d
d
d

d
422
2

The magnetic moment  = (curent) x (area of the loop)
m
Be
m
Be
44
22
2
2
d


d
 

Diamagnetism
(Classical theory)
2
2
1 4

d

m
BZeZ
i
i 
For an atom with Z electrons
If r = radius of the 3-D electron shell
2222
zyxr 
For a spherically symmetrical distribution of charge
22222
2
3
 rzyx
 is the radius of the electron loops
222
yx 

Diamagnetism
(Classical theory)
2
2
6
r
m
BZe
atom
d
 
The magnetic moment of an atom with Z electrons
Diamagnetic susceptibility per unit volume
B
N atomo
d

 
Classical Langevin equation for diamagnetism
2
2
6
r
m
NZeo
 
Number of atoms per
unit volume

Paramagnetism
• When atoms possess their own magnetic
moment, paramagnetism will occur
In a paramagnetic material the atoms contain permanent
magnetic dipole moments. An externally applied B field will
tend to align these dipole moments parallel to the field. The
result is an induced field that adds to the applied field so
that the susceptibility is positive.
1. Electron spin
2. Orbital motion of the electrons

Paramagnetism
• In atoms with filled subshells, the spin
magnetic dipole moments, and separately the
orbital magnetic dipole moments, cancel in
pairs.
• The inert gas, have closed subshell
configurations, so that they do not exhibit
paramagnetism and are excellent for
diamagnetic studies.

Paramagnetism
• Only unfilled subshells can have unpaired electrons,
so that we expect paramagnetism only in material
containing atoms whose electronic subshells are
partly filled.
• The basic requirement for paramagnetism in solids is
that the individual magnetic dipole moments have
some degree of isolation
If wave functions of atoms overlap significantly, they will tend to pair up
the magnetic dipole moments

Paramagnetism
• Unfilled inner subshells
• Isolation of the individual moments results
from the shielding of these inner subshells by
the filled outer subshells
Transition elements Rare earth elements
Paramagnetic solids

Paramagnetism
Transition elements : an element whose atom has an incomplete
d sub-shell, or which can give rise to cations with an incomplete
d sub-shell (defined by IUPAC )
Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30
Y 39 Zr 40 Nb 41
Mo
42
Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48
Lu 71 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80
Lr
103
Rf
104
Db
105
Sg
106
Bh
107
Hs
108
Mt
109
Ds
110
Rg
111
Uub
112
Rare earth elements : a collection of seventeen chemical
elements in the periodic table, namely scandium, yttrium, and
the fifteen lanthanoids (defined by IUPAC )

Paramagnetism
1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f5d<6p<7s<5f<6d<7p

Paramagnetism
• Magnetic field – line up the magnetic dipole
moments
• Thermal motion – make the directions of the
magnetic dipoles random
The susceptibility to decrease with increasing temperature
T
C
Curie law Positive constant

Paramagnetism
JgJ B  
The magnetic moment of an atom in free space is
Total angular momentum = sum of the orbital and spin
angular momentum
, gyromagnetic ratio – the ratio of the magnetic moment to
the angular momentum
B, Bohr magneton – is define as eh/4m
g – g factor or spectroscopic splitting factor

Paramagnetism
     
 12
111
1



JJ
LLSSJJ
g
The g factor comes about during the calculation of the first-
order perturbation in the energy of an atom when a weak
uniform magnetic field is applied to the system
J : the total electronic angular momentum
L : the orbital angular momentum
S : the spin angular momentum
The energy levels of the system in an applied magnetic field B
are
BE

 

Paramagnetism
When there is no field (B=0)

 NN

Paramagnetism
When an external field is applied

Paramagnetism






F
F
E
B
E
B
dEBEDN
dEBEDN




)(
2
1
)(
2
1
The total magnetic moment
B
E
N
NNM
F2
3
)(
2

  
Pauli paramagnetism

Ordered magnetic materials
• Interaction between the 3d electrons in these materials
have two effect on the magnetic moments of the ions
1. Interaction affects the orbital angular momentum
in such a way that the average orbital angular
momentum on neighbouring ions cancels. Magnetic
moment due to orbital angular momentum become
zero.
2. The spins of the 3d electrons interact in such a way
that there is a correlation between the spins of 3d
electrons on neighbouring ions. This is called the
exchange interaction.

• If the interaction between the electrons is
sufficiently strong, it is energetically
favourable for the electrons in the two ions to
have the same spin.
• Each ion affects the dipole moment of each of
its neighbouring ions, then all of the atomic
dipoles in the crystal will be aligned in a
common direction.
• Such a material is called a ferromagnet.

• The exchange interaction provides a
mechanism for aligning the atomic dipoles
without the need for an external field.
• Ferromagnet can have a large magnetization
even when no external field is present.

• Some materials retain their magnetization even in
the absence of a magnetic field.
• These materials are referred to as permanent
magnet.
• To explain this type of behaviour, require some
mechanism for causing the magnetic diople on
neighbouring ions to become mutually aligned
without the aid of an external magnetic field.
• The 3d electrons in the iron group of transistion
metals interact strongly with similar electrons on
neighbouring ions.

Three simplest type of ordering of
atomic magnetic moments .
(a) Ferromagnetic.
(adjacent magnetic
moment are
aligned)
(b) Antiferromagnetic
(adjacent
magnetic
moment are
antiparallel)
(c) Ferrimagnetic
(adjacent magnetic
moment are
antiparalle and
unequal magnitude)

Long range magnetic ordering
• Long range magnetic ordering is duo to exchange field from
neighbors
• Magnetic ordered states occurs only ay low temperatures (T <
Tc). When T > Tc there will be no ordering and the material
has to be paramagnetic
• Three common types of magnetic ordering :
Spontaneous magnetisation

• Another possible way of
ordering the dipoles is if the
magnetic dipoles in
adjacent planes are
misaligned in such a way
that the dipole form a helix.
• Example, in magnesium
dioxide the angle between
the dipoles is about 129o.
• These materials are called
helimagnets.
In a helimagnet the
magnetic dipoles are rotated
by a fixed angle

Example
Show that the magnetic susceptibility due to
the delocalized electrons in a metal is given by
.)(
2
omFm mEg  

Solution
• The number of electrons in the shaded region
on the left hand side is
• Factor of ½ is because only using one-half of
the electron distribution.
• If the total number of valence electrons per
unit volume is N, then the number aligned
parallel to the field is
• number aligned antiparallel to the field is
omF BmEg )(
2
1
.)(
2
1
2
omF BmEg
N
N 
.)(
2
1
2
omF BmEg
N
N 

Solution
• The difference is therefore
• Since each electron contributes a magnetic
moment mm.
• The magnetism is
• Susceptibility
.)( onF BmEgNN  
omFm BmEgmNNM
2
)()(  
2
)( mFo
o
o
m mEg
B
M


 

Example
the probability that an atomic dipole moment
mj is given by where A is a contant as
a function of temperature.
Show that the temperature dependence of the
magnetic susceptibility can be written as
where C is a constant.(called Curie constant)
,kT
Bm oj
Ae
T
C
m 

Solution
• If there are N magnetic ions per unit volume,
then the total number of dipole in this state is
and the magnetic moment of
these atom is
Magnetization M of the crystal
kT
Bm oj
NAe
kT
Bm
j
oj
NAem


j
j
kT
Bm
j
oj
NAemM

Solution
• Since at room temperature mjBo is very much
smaller than the thermal energy kT,
• Magnetization becomes
where C’ is a constant
kT
Bm
e ojkT
Bm oj
1
T
BC
m
kT
NAB
kT
Bm
NA
kT
Bm
mNA
kT
Bm
NAmM
o
j
j
j
o
j
j
oj
j
j
oj
j
j
j
oj
j




































2
2
2
1

Solution
• Susceptibility
T
C
T
C
B
M o
o
o
B 





Example
Calculate the magnetization of magnetite,
Fe3O4, assuming that the magnetization is due
only to the Fe2+ ions (which have six 3d
electrons) and that only the spin angular
momentum of the electrons contributes to
the magnetic moment of the ions. (The molar
volume of Fe3O4 is 4.40×10-5 m-3.)

Solution
• According to the Hund’s rule, the six 3d electrons
in each Fe2+ ion are arranged so that the spins of
five of the electrons are parallel to one another
and the sixth electron is antiparallel.
• This can be represented as ↑↑↑↑↑↓.
• Since each electron has a spin magnetic moment
of µB, the net magnetic dipole moment per ions is
4 µB, and so the dipole moment per mole is 4
µBNA where NA is avogadro’s number.

Solution
• The magnetization
.mJT1007.5
molm1040.4
)mol1002.6)(TJ1027.9)(4(
memolar volu
4
315
135
123124







 AB N
M


Temperature dependence of
permanent magnets
• When a permanent magnet is heated it is found
that there is a temperature above which the
magnetization of the material disappears.
• The transition point is known as the curie
temperature.
• Above the curie temperature the material
behaves like a paramagnet- the dipoles are
randomly aligned unless they are exposed to an
external magnetic field.

permanent magnets
• The magnetic susceptibility of this magnetic
material is given by a modified form of the Curie
law, known as Curie-Weiss law,
where c is the Curie temperature.
• As the temperature of a ferromagnet is lowered
towards the Curie point, all the atomic dipoles
move to the lowest energy state and therefore
become aligned along a common direction.
c
m
T
C





Values of the saturation magnetization at 300 K shown as
Ms (JT-1 m-3), µoMs(T) and Curie temperature c(K).
Ms (×105 J T-1 m-3) µoMs (T) c (K)
Iron 17.1 2.15 1043
Cobalt 14.0 1.76 1388
Nickel 4.85 0.61 627
Gadolinium 20.6 2.60 292
CrO2 5.18 0.65 386
Fe3O4 4.80 0.60 858
MnFe2O4 4.10 0.52 573
NiFe2O4 2.70 0.34 858

permanent magnets
• In iron, cobalt and nickel the curie temperature is
well above room temperature.
• Some material, their Curie points are below room
temperature, such materials are not normally
used as permanent magnet.
• The existence of the Curie temperature can be
explained as follow.
• As the temperature is increased, the thermal
vibrations of the ions become so large that the
alignment of the magnetic dipoles is destroyed.

permanent magnets
• This happens when the thermal energy kT is
comparable with the interaction energy between
the neighbouring dipoles.
• Since thermal effects are random in nature, we
might expect the transition to occur gradually
over a temperature range of several degrees, but
the Curie temperature is remarkably well defined.
• To understand why this is so, let us consider what
happens if we have a ferromagnetic material in
the absence of any external magnetic field.

permanent magnets
• Suppose that initially the temperature is higher than the Curie
temperature.
• In this case the magnetic dipoles are orientated in random directions.
• If the temperature is now lowered, then as we reach the Curie
temperature we find that the thermal vibrations are weak enough to allow
a few neighbouring magnetic dipoles to become aligned.
• As the result these aligned dipoles create a local magnetic field within the
sample, so other neighbouring dipoles tend to become aligned in the
same direction, which in turn further increases the strength of the internal
magnetic field.
• So we have a runaway process which does not stop until all of the
magnetic dipoles in the crystal are aligned in the same direction.
• Such process is known as a cooperative transition.
• Consequently, the change from a disordered paramagnet to an ordered
magnetic material takes place over a very narrow temperature range.

Application of magnetic materials for
information storage
• This includes audio and video tapes, floppy disks and hard
disks for computers.
• The basic principle of magnetic storage of information is
broadly similar regardless of whether the data stored is
digital or analogue.
• The magnetic medium is covered by many small magnetic
particles, each of which behaves as a single domian.
• A magnet- known as the write head- is used to record
information onto this medium by orientating the magnetic
particles along particular direction.
• Reading the information from the tape or disk is simply the
reverse of this process- the magnetic field produced by the
magnetic particles induces a magnetization in the read
head.

Schematic diagram showing the process of recording
information onto a magnetic medium
Current in
“1” “1” “1” “1”“0” “0”
B

information storage
• The choice of material for magnetic recording
medium depends on several factors.
• The coercivity must be low enough so that the
orientation of the particles can be altered by the
write head, but high enough so that the
orientation is not accidentally affected by other
external magnetic fields.
• The curie temperature should also be well above
the temperature to which the material will
normally be exposed.

information storage
• Magnetic hard disks are used in computer
because they are considerably cheaper than
semiconductor memory.
• They are not volatile.
• Magnetic system does not require a constant
electric power supply in order to remain in the
same memory state.

information storage
• In order to keep pace with the advances in
semiconductor technology it has been necessary
to increase continually the storage density of
these magnetic hard disks, from about 2 MB cm-2
in 1980 to 75 MB cm-2 in 1996.
• Firstly the size of magnetic particle is reduced.
• The read head are made using magnetoresistive
materials. ( resistance change in the present of
magnetic field)
• Align the particle so that they are perpendicular
to the surface of the disk.

Orientation of particle perpendicular
to the surface to increase its density.
Orientated parallel to
the surface
Orientated perpendicular to
the surface

information storage
• The write process is achieved by using a laser
beam to heat a magnetic particle above the Curie
temperature as a magnetic field is applied.
• As it cool down the particle retains the
magnetization of the applied field.
• To read the information another (weaker)
polarized laser beam is directed onto the surface.
• The polarization of the reflected beam is
determined by the direction of magnetization of
each particle.

Magnetism

More Related Content

What's hot

Viewers also liked

Similar to Magnetism

More from Gabriel O'Brien

Recently uploaded

Magnetism