Chapter XII Magnetic materials
Investigation of the magnetic properties of materials is very important,
because magnetic phenomena have various scientific and technical
applications.
The macroscopic properties of matter are a manifestation of the
microscopic properties of the atoms of which it is composed.
The magnetic properties of materials may be very different for types of
material, depending on their nature and structure.
GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
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Chapter XII
Magnetic materials
§1. Atomic magnetic moment - Bohr magneton
§2. Magnetization, paramagnetism and diamagnetism
§3. Ferromagnetism
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Investigation of the magnetic properties of materials is very important,
because magnetic phenomena have various scientific and technical
applications.
The macroscopic properties of matter are a manifestation of the
microscopic properties of the atoms of which it is composed.
The magnetic properties of materials may be very different for types of
material, depending on their nature and structure.
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§1. Atomic magnetic moment – Bohr magneton:
In order to understand the magnetic properties of matter we must
know the magnetic properties of atoms.
1.1 The magnetic moment of an orbiting charge :
Moving electrons, protons, neutrons create currents they have
magnetic dipole moments.
The motions of these particles can be decomposed into orbital
motion and spinning motion
The current created by the orbiting particle is
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• Its magnetic moment is
The vector form:
where L0 is angular momentum of
the orbital motion.
It is known that for an electron in the ground (non-excited) state of
the hydrogen atom the angular momentum equals to 1.05 x 10-34 J.s
(we will learn later in quantum physics), so we have for the orbital
magnetic moment of electron:
This quantity is the fundamental unit of magnetic moment, it is called
Bohr magneton 9.22 x 10-24 A.m2.
=
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2.3 The magnetic moment of a spinning charge:
Consider a spinning charge:
A ring dq of the spinning charge creates the current:
The corresponding magnetic moment of the ring is
Summing over all the rings:
Assume that the charge is distributed in the same way as the mass,
we can write
I= L
The spin magnetic moment vector
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The particular case of an electron:
An electron is known to have spin angular momentum of
0.527 x 10-34 J.s. So, its spin magnetic moment is
However, experiments give the result of twice bigger. Why?
The problem lies with the assumption about the charge distribution
that we have used.
To correct this mistake one introduces the factor g called
“gyromagnetic ratio“, and writes
For electrons g = 2.
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§2. Magnetization, paramagnetism and diamagnetisme:
2.1 Some general view on the magnetic properties of materials:
The material which has the most striking magnetic properties is
iron. Similar magnetic properties are shared also by nickel,
cobalt,…. That kind of magnetic properties is called
ferromagnetism.
All other ordinary substances do show some magnetic effects,
but very small ones – a thousand → million times less than the
effects in ferromagnetic materials.
This small magnetism is of two kinds. In other words, there are
two signs to the magnetic effects: paramagnetism and
diamagnetisme.
Strong magnetic effects Weak magnetic effects
in ferromagnetic materials in paramagnetic in diamagnetic
materials materials
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Two signs to the magnetic effect:
• If the small cylinder is of bismuth → it is repelled by the sharp pole
• If the small cylinder is of aliminium → it is attracted by the sharp pole
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2.2 Magnetization:
First we consider a piece of paramagnetic
material.
When no external magnetic field is present, the
atomic magnetic dipoles are randomly aligned
(pic. a). The total magnetic field due to all the
dipoles cancels to zero.
If we apply an external magnetic field B 0 (pic. b),
the dipoles tend to align with the applied field, and a)
the vector sum of all atomic magnetic moments
becomes non-zero vector, then we denote it by B’ :
Applying the formula of magnetic field for a ring
current on its axis, we have for each atomic current:
b)
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Since atomic current ring is very small, we have
where we have introduced the quantity M , considered as a type
of average dipole moment per unit volume. We can write
and we obtain the total magnetic field is
The quantity M is called the magnetization of the material.
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Units:
The units of magnetic moment are (current) x (area), that is A.m 2
The units of magnetization M is (A / m2) / m3 = A / m
The units of M is the same as the units of B (as it must be):
( T.m / A )( A / m ) = T
It is natural to think that the sum of the atomic magnetic moments tend to
align with the external magnetic field, then
B > B0 and M > 0
However, by experiments one observed that this is true not for all
materials, but only for most common materials. Such materials are called
paramagnetic. For these materials M > 0 but fairly small.
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2.3 Permeability and susceptibility:
We have said that for paramagnetic materials the total magnetic field
inside the material B is greater than the external field B0 . So we can write
B = Km B0
where Km is a dimensionless factor, called relative permeability.
The value of Km is typically ranges from 1.0001 → 1.003 (see the table
In the page 1089 of the textbook).
The expression of the magnetic field in materials relates to that in
vacuum by the replacement by
Km
=
which is called the permeability of the material.
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Since for paramagnetic materials the value of Km is a small deviation
from unity, it is convenient to introduce the quantity
χ = Km – 1
m
which is called the magnetic susceptibility.
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2.4 Diamagnetism:
There are a type of materials for which the
magnetization vector is opposite to the external
magnetic field, that is M < 0.
Why? We can explain this phenomenon in
a simplified version as follows:
a)
In the absence of external fields the
electrons move randomly (pic. a).
When the external magetic field is applied (pic. b),
the electrons begin move in circular orbits. This
orbiting electrons create a field which is opposite
to the external field.
This type of materials is called diamagnetic. For it
Km is typically of the order of 0.9999 → 0.99999
( see the table). b)
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§3. Ferromagnetism:
3.1 Strong magnetization of ferromagnetic materials:
The third type of materials is called
ferromagnetic materials, which includes
iron, nickel, cobalt, … These materials
manifest strong magnetic effects.
The magnetic field inside them is much
larger than the applied external field, the
relative permeability Km is of the order of
1.000 to 10.000.
The properties of ferromagnetic materials
are explained by their microscopic structure:
In these materials the atomic magnetic moments are extremely
easy to align together, due to strong interactions between them.
Inside the material there exist regions called magnetic domains,
even when no external field is applied. In each domain the atomic
magnetic moments are parallel to each other.
In the absence of external magnetic field the magnetic moments
4/1/2008 of different domains align randomly as shown in the picture. 16
When an external field B0 is applied, the domain magnetic moments
tend to orient themselves paralell to the field that leads to the shift of
domain boundaries: domains that have magnetic moments parallel to
the external field will grow, other domains will shrink.
The magnetic moment of each domain have the order of thousands
of Bohr magnetons, the torques that tend to align the domains with the
external field are much stronger than in paramagnetic materials. After
rearrangment of domain magnetic moments the magnitude of the
magnetization vector of the material is much larger than the external field.
Two important features of ferromagnetism are
• the saturation of magnetization
• the hysteresis
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3.2 Saturation of magnetization and hysteresis:
Consider the dependence of the magnitude of the magnetization vector
on the magnitude of external magnetic field.
Increase B 0 from zero → the magnitude
M of magetization increases.
Msat When B0 reachs to some enough large
value, further increase of the external field
causes no increase in magnetization. This
phenomenon is called the saturation of
magnetization.
B0
The saturation of magnetization is explained as follows: When the
external field is enough large, all the domain magnetic moments are
aligned parallel to it, and the magnetization can’t increase further.
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M
When the material is magnetized to
saturation we reduce the external field
to zero, the magnetization decreases
(the curve b), but some magnetization
remains when B0 = 0.
The material becomes then a
permanent magnet. It has own
magnetic moment when the external B0
field is removed.
To reduce the magnetization to zero
We must apply an external field in the
inverse dirction.
The variation of the magnetization with Hysteresis loop
the change of the applied magnetic
field is described by the hysteresis loop.
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