Lesson 22
(1) Microscopic magnetic moments
Motion of charged particles inside atoms gives rise to microscopic magnetic
moments. These are related to the angular
momentum of the particle.
For a particle with momentum p = mu , the
angular momentum about a point O is
L=r´p
where r is the vector pointing from O to the
position of the particle.
For a particle travelling with constant speed
u in a circle of radius r , the magnitude of the angular momentum about the
center of the circle is
L = mru
The direction is the same as the direction of advance
of a right hand screw turned in the direction of the
velocity. The frequency of the motion is
f=
u
2p r
If the particle carries the charge q , the circular motion gives rise to a current
I = qf =
qu
2p r
and a magnetic moment
m = p r2I =
qu r qmru
q
=
=
L
2
2m
2m
Taking the direction into account, we find
m=
q
L
2m
If the charged particle is electron, the magnetic moment is
1
mL = -
e
L
2me
In quantum mechanics, the angular momentum is found to be an integral
multiple of Planck constant , which has the value
=
h
=1.05´10-34 J × s
2p
Expressing angular momentum is units of
m L = -m B
, we can write
L
after introducing the Bohr magneton
mB =
e
= 9.27 ´10-24 A × m 2 = 9.27 ´10-24 J / T = 5.79 ´10-5 eV / T
2me
The Bohr magneton gives an estimate of the size of microscopic magnetic
moment due to electrons.
An electron also has angular momentum due to its intrinsic spin that is unrelated
to motion. The spin angular momentum S gives rise to magnetic moment
mS = -2
e S
S
= -2m B
2me
Although the factor for m S is twice that for m L , the spin angular momentum S
assumes half-integral values instead of integral values as for orbital angular
momentum L .
(2) Magnetization
By definition, the magnetization of a material is the magnetic moment in unit
volume of the material. The magnetic moment is the vector sum of the atomic
magnetic moments. In most substances, the atomic magnetic moments are
randomly orientated, so that the magnetization is zero. If all atomic magnetic
moments are aligned, we find the maximum magnetization called saturation
magnetization.
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Example: Find the saturation magnetization of iron, assuming that each atom has
the magnetic moment of 1 Bohr magneton.
Solution: Density of iron = 7.87´103 kg / m3
Molar mass of iron = 55.8g / mole = 55.8´10-3 kg / mole
Number density of atoms
7.87 ´103 kg / m3
n=
´ 6.02 ´10 23 atoms / mole = 8.49 ´10 28 atoms / m3
-3
55.8´10 kg / mole
Saturation magnetization
M sat = nmB = 8.49 ´1028 atoms / m3 ´ 9.27´10-24 A× m = 7.88´105 A / m
Magnetization of a material is associated with macroscopic current and magnetic
field. To find the current, consider a slab of material with area A and thickness
L . Consider the atomic currents giving rise to the atomic moments to be square
loops of current i of area a , filling up the whole area A , and that there are N L
layers of such loops in the
thickness L .
Number of loops in a layer N A =
A
a
Magnetization =
åm = N
V
N Aia N Li
=
LA
L
L
Adding up these loop currents, we see that there is no current in the interior, but
there is a current
I = N Li
running on the edges of the slab called the Amperean current. This behaves like
a solenoid current and creates a magnetic field in the interior equal to
BM = m0
NL
I
i = m0 = m0 M
L
L
Example: For the sample of iron in the previous example, the magnetic field due
to maximum magnetization is
B = m0 M sat = 4p ´10-7 ´ 7.88´105 = 0.99T
(3) Induced magnetization
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The magnetization of most substances is zero. It can become non-zero when an
external magnetic field Bapp is applied, such as by placing the sample inside a
solenoid. This is called induced magnetization. For paramagnetic and
diamagnetic substances, the induced magnetization is proportional to the
applied field, so that we can write
M = cm
Bapp
m0
where the material-dependent constant c m is called magnetic susceptibility. It is
positive for a paramagnetic substance and negative for a diamagnetic substance.
Since the total magnetic field inside the sample is
B = Bapp + m0 M = (1+ c m ) Bapp
it is enhanced in a paramagnetic substance and reduced in a diamagnetic
substance. The values of susceptibility for most substances are quite small.
When an inhomogeneous external magnetic field is applied to the sample, it is
attracted to the region of strong field is it is paramagnetic, and expelled from
that region if it is diamagnetic. This can be explained by the fact that the induced
magnetization is parallel to the applied field for paramagnetic and anti-parallel
fro diamagnetic materials, if we also remember that a magnetic moment behaves
like a small bar magnet. The explanation is illustrated in the diagrams shown.
However, the forces involved are weak.
(4) Paramagnetism
In a paramagnetic substance, the atoms have permanent magnetic moments.
Because of thermal motion, these magnetic moments are randomly orientated.
They are aligned to some degree when an external magnetic field is applied,
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giving rise to magnetization. We can estimate this magnetization in the following
manner.
Magnetic energy of atom ~ matom Bapp
Random thermal energy of atom ~ kT
where T = temperature in K, k = Boltzmanncons tant =1.38´10-23 J / K
Since the applied field counteract randomization due to thermal motion, the
fraction of magnetization versus saturation magnetization can be estimated to
be
m B
m B
M
M
~ atom app , with
= atom app in an exact calculation
M sat
kT
M sat
3kT
This leads to the magnetic susceptibility
cm =
matomm0 M sat
3kT
which is known as Curie’s law. According to this, susceptibility decreases with
temperature.
As an example, aluminum is paramagnetic, with susceptibility c m = 2.2 ´10-5 at
20 C .
(5) Ferromagnetism
In iron, cobalt, and a few other metals, because of the strong forces between the
electron spins in neighboring atoms, the spins are lined up in the same direction,
making large magnetic moments. These magnetic moments exist in domains, but
the directions of the magnetic moment in the domains are still random, so that
magnetization is still zero ordinarily.
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When a magnetic field is applied, the domains can slip against each other,
causing their magnetic moments be aligned with the applied field. Much larger
induced magnetization occurs compared with paramagnetic substances. The
substance is called ferromagnetic.
As the applied field increases, the
magnetization also increases,
proportionately to the applied
field when the latter is small, but
eventually more slowly and
approaches saturation, as shown
in a plot of M against Bapp . If the
applied field is now reduced, so
does the magnetization. However,
the curve of M against Bapp does
not retrace the original path. In
particular, when the applied field
is reduced to zero, magnetization does mot go back to zero, and takes on a finite
value. The material now becomes a permanent magnet. Increasing the applied
field in the opposite direction can eventually reduce the magnetization to zero,
(demagnetization) and further increase leads to saturation in the opposite
direction. Increasing Bapp back in the original direction traces out a lower branch
of the plot of M against Bapp . Thus, magnetization does not depend simply on
the applied field. It also depends on which branch of the curve of M against Bapp ,
and therefore past history the sample has gone through. Such phenomenon is
known as hysteresis. When the substance is brought through a hysteresis loop,
work is done by the applied field, which is turned into heat. The work is
proportional to the area inside the hysteresis loop.
Above a temperature known as Curie temperature, the substance ceases to be
ferromagnetic. For iron, this temperature is 770°C.
We can still define susceptibility for a ferromagnetic substance by
cm =
m0 M
Bapp
except that c m now depends on Bapp and past history. If the substance is placed
inside a solenoid with turn density n and current I , so that the applied field is
Bapp = m0 nI ,, the magnetic field inside the substance is
B = Bapp + m0 M = (1+ c m ) Bapp = (1+ c m ) m0 nI
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m = (1+ c m ) m0
Defining magnetic permeability by
B = mnI
we can write the magnetic field inside the solenoid as
Because of the similarity of this to the vacuum case, m 0 is called the permeability
of free space. The relative permeability m m0 can be as large a few thousands,
enabling the construction of strong electromagnets.
(6) Diamagnetism
If a substance contains free charged particles, they will
describe circular motion when a magnetic field is applied.
The circular motion generates magnetic moment.
Considerations show that the magnetic moment is opposite
to the direction of the applied field, whatever the sign of the
charge. The susceptibility is then negative, and the
substance is called diamagnetic. An example of such
substance is high temperature plasmas, which are ionized
gases made up of free electrons and positive ions.
In most materials, many electrons are bound to individual
atoms, travelling around the nucleus in closed orbits. Because of their random
orientation, the magnetic moments due to these orbits cancel out, and no
magnetization exists. When a magnetic field is applied, these orbits undergo
precessions, all in the same direction. The magnetic moments associated with
precessions now add up to give non-zero magnetization. The direction of this
magnetization can be shown to be opposite to the applied field. The material
then exhibits diamagnetism. The mechanism of diamagnetism is present in all
substances. Since accordingly to Curie’s law, the magnetic susceptibility of
paramagnetic substances goes down as temperature increases, all substance
should become diamagnetic at high enough temperature.
(7) Superconductor
A superconductor not only has zero
electrical resistivity, it is also a perfectly
diamagnetic substance. The magnetic
susceptibility is c m = -1. Therefore
when it is placed in an external
magnetic field, the latter cannot
penetrate. The field lines near a
superconductor show such exclusion.
There is surface current on the
conductor that generates its own field in
the interior to cancel out the external
field . This is called the Meisner effect.
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