MAGNETIC PROPERTIES OF MATERIALS
Background
Before considering magnetism as related to multi-atomic system, we need to solve the problems on an atomic
scale through the wave aspects of particles (wave mechanics) and this has been achieved by Schrödinger with his
wave equation which in time dependent form is
h 2   2  2  2
 2  2
8 2 m  x 2
y
z

  h   V
 2j t

(3-1)
where V is the potential energy and  is the amplified of the wave function. To obtain V, we consider the
hydrogen atom and use spherical co-ordinates (r, , ) since the potential has a spherical symmetry, then
V 
e2
4 o r
Equation (3-1) becomes
 2 r , ,  
2m 
e 2 
E
 r , ,   0
2 
4 o r 
 
(3-2)
h 

 

2 

In polar co-ordinates the full expression is
1   2  
1
 
 
1
 2
(3-3)
 2  2
r
 2
 Sin
 2 2
r  r Sin  
  r Sin   2
r r 
This equation can be solved if  is expressed as a function of 3 variables r, ,  and these can be solved
separately to obtain
 r , ,    Rr   F      
Each of these functions R(r), F(), () mirrors a motion of the electron in terms of the corresponding coordinate. Each solution throws up its corresponding quantum number n, l, ml, ms:
 n which is any positive integer (called principal or radial quantum number). It describes the energy of a
given state. States for which n = 0, 1, 2, 3, are called K, L, M, N – shells and the maximum number of
electrons that can be contained in states characterised by the principal quantum number n = 1, 2, 3,….
are equal to 2, 8, 18, …..2n2 electrons.
 “l” which has values 0, 1, …. (n-1) (called orbital or azimuthal quantum number), measures the
component lh of angular momentum number). The higher the angular momentum quantum number, the
lower the probability of the electron being near the nucleus, and vice versa. It determines the geometric
characteristics (or shape) of the electron probability distribution. States for which l = 0, 1, 2, 3, 4, 5 are
called s, p, d, f, g, h…. subshells. Therefore for a given value of n, the maximum numbers of electrons
that can be contained in s, p, d and f states are 2, 6, 10, and 14 respectively, which is 2(2l+1) electrons.
 ml which has values –l, -(l –1), … 0, …. (l – 1), l (called the magnetic quantum number), measures the
components mlh of the angular momentum. Describes the orientation of the electron orbital magnetic
field with respect to an applied field. This number determines the orientation in space of the electron
probability distribution. Since m affects the energy of the electrons only when they are in an applied
field, in the absence of a field, electrons having different ml values may still have the same energy.
 s which is the last quantum number is due to the “spin” of the electron about its own axis and it is
therefore called the spin quantum number “s” and it can only have 2 components ms = 1/2
corresponding to angular momentum 1 2  (  being the basic quantized unit). This is the quantum
number that identifies the electrons in a state as being either "spin up" or "spin down". This quantum
number is very important for determining magnetic effects in matter.
where
The Pauli Exclusion Principle states that no two electrons may occupy the same energy state in an atom. This
means that no two electrons may have the same set of values for the quantum numbers n, l, m, s because they
would then be indistinguishable. As electrons are added, they fill up each possible state in a given subshell in
parallel spin configurations first (HUND'S RULE) before filling the shell associated with the next higher energy
state. The filling of the shells is governed by Schrödinger's wave equation and the quantum numbers.
3s
Na
3p
3d
4s
Sc
Ti
Mg
E
4f
4d
4
3
V
Al
4p
3d
4s
3p
Si
3s
2p
P
2s
S
2
Cr
Mn
Fe
Co
Ni
1
1s
Cl
Ar
Cu
Zn
Figure 3-1:
If all electrons are paired, there is no "spin" magnetic moment. These materials are still magnetic though, due to
the electron's orbital motion. The spin structure of the transition series elements (iron in particular) is most
important for magnetic biomaterials. This is due to the uncompensated spins in the 3d orbital. This gives rise to a
"spin" magnetic moment. The spin moment is much stronger than the orbital moment and is aligned parallel to
an applied field.
Atomic Magnetism
The orbital and spin magnetic moment of an electron are a consequence of the orbital and spin angular
momentum. In terms of the appropriate quantum number,
 eh 
 orb  ml 
(3-4)

 4m 
 eh 
(3-5)

 4m 
where the factor g is a multiplying factor which is found to be necessary to bring the relation for spin moment
into line with that for orbital moment. It has the value of almost 2 and is called the Lande splitting factor. Since
ml is an integer, the quantity (eh/4m) is regarded as a natural unit of magnetic moment, called the “Bohr
magneton, B”. It has the value 9.27 x 10-24 amp.m2.
To obtain the total magnetic moment of an atom, all the magnetic moments of the electrons in the atom must be
added up taking into consideration the fact that the direction of the moments is given by the sign of ml and ms. In
this way, closed shells of electrons having equal numbers of electrons with positive and negative values of mI
and ms have no net magnetic moment since each electron cancels another.
 spin   gms 
Example
The atomic number of neon is 10 giving rise to electronic configuration 1s2 2s2 2p6. The dipole moment of each electron for the p
state is represented by an arrow in Table 3.1.
Therefore the first important rule to note is that a net magnetic moment comes only from incomplete electron
shells. In transition elements, one of the inner electron shells (3d shell) is incomplete and this suggests
immediately that the inner incomplete shell is the one which gives rise to the atomic magnetic moment while the
valence electrons are of no importance. This characteristic of the valence electrons is just a reflection of the
properties of atomic bonding since atoms bond together in such a way as to form closed shells of electrons which
have no magnetic moment.
Table 3.1:
Electron
Moment due to ml
ms
Moment de to ms
1

+1/2

2

-1/2

3
+1/2

4
-1/2

5

+1/2

6

-1/2

Magnetic induction B
Generally, in the presence of a magnetic field in vacuum, the magnetic induction or the magnetic flux density B
is given by
B  o H
(o is the permeability of vacuum)
(3-6)
When a magnetic material is in a magnetic field H with a magnetization (or dipole moment/unit volume) M, the
magnetic induction B in the material is
B   o H  M    o  r H
(3-7)
wherer = relative permeability of the material (r = 1 for vacuum), hence
M 
r  1     1 
(3-8)
H
where is the magnetic susceptibility of the material, B is measured in Tesla or weber.m-2, M and H are
measured in ampere m-2 , so  has no unit. The following figure shows the effect of introducing a magnetic
material in a magnetic field compared to when the magnetic field is acting in a vacuum. Compare the graph with
the effect of an electric field on the polarisation of a dielectric material.
Magnetic field H
Figure 3-2:
Classification of Magnetic Materials
There are five classes into which magnetic materials may be grouped.
1.
Diamagnetic
2.
Paramagnetic
3.
Ferromagnetic
4.
Anti-ferromagnetic and
5.
Ferrimagnetic
Diamagnetism
Materials in which all electron spins are paired (i.e. there are no uncompensated spins).Diamagnetism is a
property of all atoms because of the influence of an applied magnetic field on the motion of electrons in their
orbit. It is a very weak effect and in solids, it is often masked by other kind of magnetism.
O
+e
w
-e

Figure 3-3:
Consider an electron moving round a proton in an orbit of radius r (as in the figure above) with angular
frequency ‘w’ in the absence of a magnetic field. There is a magnetic moment  associated with the motion of
the electron given by
 = current x area
 e 
  . 2
 2 
(3-9)
With the application of a magnetic field H, there is a flux change through the orbit and by Lenz’s law an electric
field is set up to oppose this change. This happens only when the magnetic field is changing from its zero value
to the steady value H. The electric field affects the motion of the electron by exerting a force on it, which
changes its angular momentum. The change in the angular frequency this effected is called the “Lamor
frequency” and has the value
eB

(3-10)
2m
where B is the flux density appropriate to H and m is the electron mass. The corresponding induced magnetic
moment is opposed to the original magnetic moment  of the electron in its orbit when H = 0. For a solid with
atomic number Z, the expression for the magnetic moment “”, after substituting for w in the equation 10
becomes
Ze 2 B 2
 2  x2  y2
4m
where  2 is the mean square of the perpendicular distance of the electron from the field axis through the nucleus.



The mean square distance of electron from the nucleus is r 2  x 2  y 2  z 2 and for a spherically symmetrical
distribution of charge.
so that r 2  32  2
x2  y2  z2
Therefore, the diamagnetic susceptibility per unit volume is if N is the number of atom per unit volume given by

 o N

 o Ne 2 r 2
(3-11)
B
6m
This is the classical Langeuim result and it shows that diamagnetic materials exhibit negative susceptibility.
Typical Experimental values of the molar susceptibilities are the following:
Table 3.2:
He
-1.9
Ne
-7.2
Ar
-19.4
Kr
-28.0
Xe
-43.0
As a summary, a diamagnetic material is a substance with a negative magnetic susceptibility.
Paramagnetism
Materials in which there are uncompensated spins (i.e. there is not a spin -1/2 for every spin +1/2). Each electron
in an orbit has an orbital magnetic moment orb and a spin magnetic moment spin. In atoms with filled electronic
shells, there is no net magnetic moment but when the shells are unfilled, there is such a moment. In the absence
of a magnetic field, the net moments of the atoms usually point in random directions because of thermal
fluctuations, producing no net macroscopic magnetization. With a field switched on, there is a tendency for the
atoms to align with the field giving an induced positive magnetic moment, which is proportional to the applied
field. The induced magnetization is the source of paramagnetism and materials whether gases, liquids or solids
show a paramagnetic susceptibility whenever the paramagnetic atoms can respond individually to an external
field without interaction with their neighbours. Quantum mechanics governs the component of each individual
electronic moment in the field direction.
In general, solids exhibit paramagnetism if they contain paramagnetic atoms in dilute solution, i.e., one such
atom or ion must be far enough away from another, with diamagnetic atoms in between, so that there is no direct
magnetic interaction between them.
Quantum Theory of Paramagnetism
The magnetic moment of an atom or ion in free space is given by
  J   g B J
where the total angular momentum J is the sum of orbital L and spin S angular momenta. " " , which is
called the gyro-magnetic ratio or magnetogyric ratio, is the ratio of the magnetic moment to the angular
momentum. For electronic system, a quantity “g” called the g-factor or the spectroscopic splitting factor is
defined by
(g is usually taken to be 2)
g B  
For a free atom the g – factor is given by the Lande equation
1  J J  1  S S  1  LL  1
(3-12)
g
2 J J  1
The total magnetic moment of an atom or ion is
(3-13)
   B L  2 S 
If we can again employing the terminology of Lande (1923), this moment can then be written as
(with g given in equation 3-12)
(3-14)
  g B J
For each value of m j , the total magnetic moment  of equation (14) has a component of g B m j aligned with the
field axis. Such a component of magnetic moment aligned with the field axis requires a potential energy
U l , m   g B m j B     B
This is equal to the energy levels of the system in a magnetic field. B is called the “Bohr magneton” and is equal
to the spin magnetic moment of a free electron.
e
B 
2m
m j is the azimuthal quantum number and has the value J, J – l, …..J. For a single spin with no orbital moment,
we have m j  1 2 and g = 2 which gives U    B B . This splitting is shown in the figure below:
Figure 3-4:
If a system has only 2 levels, the equilibrium population following Boltzman’s distribution is given by
N1

N
e
B
e
B

 B
  k BT 
(3-15)

e
B
N2

N
e
B
e

(3-16)
 B

e
N1, N2 are the populations of the lower and upper levels with N = N1 + N2 being the total number of atoms. The
projection of the magnetic moment of the upper state along the field direction is - and the lower state is . The
resultant magnetization for N atoms/unit volume is therefore
M  N1  N 2   N 
For x << 1, tanh x
e x  ex
e x  e x
 N tanh x
where x 
B
k BT
(3-17)
 x, then we have
N 2 B
k BT
In a magnetic field, an atom with angular momentum quantum number J has 2J + 1 equally spaced energy level.
The magnetization is usually given by:
M 
M  NgJ B BJ x 
x is a dimensionless variable given by "
(3-18)
gJ B B
" and BJ(x) is the Brillouin function expresses as
k BT
 2J  1 
 2 J  1x  1
 x 
B J x   
Ctnh
Ctnh


 2J 
 2J  2J
 2J 
1 x x3
 
 ..........
x 3 45
The Brillouin function varies from zero when the applied field is zero to unity for infinite field. Thus the
saturation magnetization of a paramagnetic solid is M max  Ng B J . Under weak field conditions, the Brillouin
function is asymptotic and the paramagnetic susceptibility has a Curie-Law Behaviour
for x << 1, Ctnh  x 

M  o M NJ J  1g 2  B2  o


H
B
3k BT

Here p  g J  J  1
1
2
(3-19)
Np 2  B2  o C

3k BT
T
is the effective number of Bohr magneton. C is the Curie constant and is given by
C
Np 2  B2  o
NJ J  1g 2  B2  o

3k B
3k B
(3-20)
Ferromagnetism
Materials in which there are uncompensated spins which are quantum mechanically coupled. A ferromagnet has
a spontaneous magnetic moment – a magnetic moment even in zero applied fields. A spontaneous moment
suggests that electron spins and magnetic moment are arranged in a regular manner. From these, it is apparent
that ferromagnetism must involve the co-operative alignment of permanent atomic dipoles, which arise in atoms
having unpaired electrons. The strength of each individual dipole, , is a small number of Bohr magnetons (B),
but a completely ordered array of such moments produces a large spontaneous magnetization, Ms. In
ferromagnetic materials, the magneton number” (/B) is determined almost entirely by the spin of unpaired
electrons, with a minor correction from orbital motion considerations.
In the simplest kind of approach of ferromagnetism, consider a paramagnet with a concentration of N ions of
spin ‘s’. If there is an internal interaction tending to line up the magnetic moments parallel to each other, then
we shall have a ferromagnet. Let us call this interaction an “exchange field” (This is also called the molecular
field or the effective field). The orienting effect of the exchange field is opposed by thermal agitation and at
elevated temperature, the spin order is destroyed. The magnitude of this exchange field BE, which may be as high
as 107 Gauss is proportional to the magnetization M
(3-21)
BE  M
M is the magnetization per unit volume and  is a constant independent of temperature. The Curie Temperature
Tc is the temperature above which the spontaneous magnetization vanishes. It separates the disordered
paramagnetic phase at T>Tc from the ordered ferromagnetic phase at T<Tc. Above Tc, the material becomes
paramagnetic and an external field must be applied to produce any magnetization, but the magnetization is then
self assisted by the exchange field it generates. In a weak applied field
(3-22)
 o M   p B a  B E 
where Ba is the applied field and p = C/T is the paramagnetic susceptibility. Substituting equation 21 in 22 gives
for T > Tc
(3-23)
M   p H  H E 
This equation has taken into consideration, the fact that
M oM


H
B
However p in equation 3-23 will be smaller than the observable susceptibility, which is
M
(the magnetization/unit applied field)
m 
H
Equations 3-21 to 3-24 can be combined to give
C
m 
T  C 
(3-24)
C
for T > Tc (where Tc = C)
(3-25)
T  TC
This is called Curie-Weiss law for susceptibility above the Curie points. The mean distance  can be determined
by considering equation 19.
T
3k B TC
(3-26)
 C 
2
C
Ng S S  1 B2  o2



Magnetic Domains
Ferromagnetic materials get their magnetic properties not only because their atoms carry a magnetic moment but
also because the material is made up of small regions known as magnetic domains. In each domain, all of the
atomic dipoles are coupled together in a preferential direction. This alignment develops as the material develops
its crystalline structure during solidification from the molten state. Magnetic domains can be detected using
Magnetic Force Microscopy (MFM) and images of the domains like the one shown below can be constructed.
Figure 3-5:Magnetic Force Microscopy (MFM) image showing the magnetic domains in a piece of heat treated carbon steel.
During solidification a trillion or more atom moments are aligned parallel so that the magnetic force within the
domain is strong in one direction. Ferromagnetic materials are said to be characterized by "spontaneous
magnetization" since they obtain saturation magnetization in each of the domains without an external magnetic
field being applied. Even though the domains are magnetically saturated, the bulk material may not show any
signs of magnetism because the domains develop themselves are randomly oriented relative to each other.
Ferromagnetic materials become magnetized when the magnetic domains within the material are aligned. This
can be done my placing the material in a strong external magnetic field or by passes electrical current through the
material. Some or all of the domains can become aligned. The more the number domains that are aligned, the
stronger the magnetic field in the material. When all of the domains are aligned, the material is said to be
magnetically saturated. When a material is magnetically saturated, no additional amount of external
magnetization force will cause an increase in its internal level of magnetization.
The Hysteresis Loop and Magnetic Properties
A great deal of information can be learned about the magnetic properties of a material by studying its hysteresis
loop. A hysteresis loop shows the relationship between the induced magnetic flux density B and the magnetizing
force H. It is often referred to as the B-H loop. An example of hysteresis loop is shown below.
B Flux density
aSaturation
Retentivity
b
Figure 3-6:
Effect of applied magnetic field on
magnetic induction B leading to
hysteresis loop.
Coercivity
-
Saturation in
d
opposite direction
c
O
e
-B
f
Magnetic field
The loop is generated by measuring the magnetic flux B of a ferromagnetic material while the magnetizing force
H is changed. A ferromagnetic material that has never been previously magnetized or has been thoroughly
demagnetized will follow the dashed line as H is increased. As the line demonstrates, the greater the amount of
current applied (H+), the stronger the magnetic field in the component (B+). At point "a" almost all of the
magnetic domains are aligned and an additional increase in the magnetizing force will produce very little
increase in magnetic flux. The material has reached the point of magnetic saturation. When H is reduced back
down to zero, the curve will move from point "a" to point "b." At this point, it can be seen that some magnetic
flux remains in the material even though the magnetizing force is zero. This is referred to as the point of
retentivity on the graph and indicates the remanence or level of residual magnetism in the material. (Some of the
magnetic domains remain aligned but some have lost there alignment.) As the magnetizing force is reversed, the
curve moves to point "c", where the flux has been reduced to zero. This is called the point of coercivity on the
curve. (The reversed magnetizing force has flipped enough of the domains so that the net flux within the material
is zero.) The force required to remove the residual magnetism from the material, is called the coercive force or
coercivity of the material.
As the magnetizing force is increased in the negative direction, the material will again become magnetically
saturated but in the opposite direction (point "d"). Reducing H to zero brings the curve to point "e." It will have a
level of residual magnetism equal to that achieved in the other direction. Increasing H back in the positive
direction will return B to zero. Notice that the curve did not return to the origin of the graph because some force
is required to remove the residual magnetism. The curve will take a different path form point "f" back the
saturation point where it with complete the loop.
From the hysteresis loop, a number of primary magnetic properties of a material can be determined.
1.
2.
3.
4.
5.
Retentivityor remanence - A measure of the residual flux density corresponding to the saturation
induction of a magnetic material. In other words, it is a material's ability to retain a certain amount of
residual magnetic field when the magnetizing force is removed after achieving saturation. (The value of
B at point B on the hysteresis curve.)
Residual Magnetism or Residual Flux - the magnetic flux density that remains in a material when the
magnetizing force is zero. Note that residual magnetism and retentivity are the same when the material
has been magnetized to the saturation point. However, the level of residual magnetism may be lower
than the retentivity value when the magnetizing force did not reach the saturation level.
Coercive Force - The amount of reverse magnetic field which must be applied to a magnetic material to
make the magnetic flux return to zero. (The value of H at point C on the hysteresis curve.)
Permeability, - A property of a material that describes the ease with which a magnetic flux is
established in the component.
Reluctance - Is the opposition that a ferromagnetic material shows to the establishment of a magnetic
field. Reluctance is analogous to the resistance in an electrical circuit.
Rectangular Loop Ferrites
Some materials have hysteresis loop switch are almost rectangular in shape. This property makes them suitable
for use in a magnetic memory core. The principle is as follows; the points Po and P1 represent two stable states of
magnetization. These states can represent a zero and one in digital storage of information. The loop can only be
traversed in an anti-clockwise direction and the state Po can be changed rapidly to P1 by the application of a field
greater than HA. Similarly, P1 can be changed to Po by the application of a field less than or equal to minus HD.
The switching is about one microsecond and it is an important parameter.
B
Figure 3-7:
Typical example of rectangular or square shaped hysteresis
loop most useful for information storage
C
D
E
P1
B
O
A
F
Po
H
Effect of Temperature
Ferromagnetism is a temperature dependent phenomenon. In fact ferromagnetism and paramagnetism are at
different ends of a thermal energy / magnetic energy scale. At a low enough temperature paramagnetic behaviour
can become ferromagnetic due to the lack of randomising thermal energy. Likewise at a high enough temperature
ferromagnets can become paramagnetic due to thermal energy randomising the direction of individual ionic
dipoles. The temperature at which ferromagnetism breaks down is called the Neel or Curie temperature and is
dependent on the material composition. Indeed it is often used to identify magnetic minerals.
Magnetic Grain Size
Magnetic crystal grain size is, as with temperature, a balance between opposing energies. For ferromagnetic
minerals there are four different balancing points that are dependent on the size of magnetic crystals. For very
small ferromagnetic crystals the effect of thermal randomisation is stronger than for larger crystals. This leads to
a phenomenon called super paramagnetism. As the name suggests these very small minerals exhibit strong
paramagnetic properties, and cannot retain remanence.
Super viscous materials are in a grain size between supper paramagnetic and single domain, they can hold
remanence but will lose it over a short period of time, the exact amount of time depends on the grain size. Super
paramagnetic and super viscous minerals both exhibit a property called frequency dependent susceptibility.
When a frequency dependent sample is subjected to a high frequency field there is a time lag between the
maximum in the field strength and the reaction of the sample, this leads to super paramagnetic particles having a
lower susceptibility when measured in high frequency fields.
The next size class up is the single stable domain crystal. In these grains the thermal energy is overcome by the
magnetic energy of larger volume crystals and a stable magnetic moment results. When the size is increased
again the crystal continues to try and minimise its magnetic energy. In large crystals this can be accomplished by
the division of the magnetic moment into two or more magnetic domains. These domains will substantially
reduce the overall magnetic moment of the crystal, and in many cases take it to zero. These large crystals are
referred to as multidomain grains.
The relative "hardness" of ferromagnetic minerals can be linked to their magnetic crystal size. A sample
consisting of predominantly single domain crystals will be relatively hard and requires high fields to influence its
magnetic remanence. Whereas a sample containing mostly multidomain grains will be magnetically soft and it
will be easy to impart a remnant magnetisation. There is also an intermediate state where most of the magnetic
crystals have only two or three domains. Intermediate hardness crystals are sometimes called pseudo single
domain crystals as they have many properties that are somewhat similar to single domain crystals, but lower
magnetic intensities.
Isothermal Remanent Magnetisation (IRM)
The IRM of a sample is the magnetisation retained by that sample when it has been subjected to a known field at
a known temperature (usually room temperature). IRM can be measured at varying fields, typically between
20mT and 3T. By increasing the field that a sample is subjected to in stages and measuring between each stage, it
is possible to gain a lot of information about a sample. The saturation IRM or SIRM is the maximum remanence
that a sample can acquire by IRM magnetisation. SIRM alone can be very indicative of a materials composition,
for instance Magnetite will usually reach saturation at approximately 300mT where as Haematite is often still
unsaturated at applied fields of 2T or 3T. However the overall shape of the IRM curve (the IRM's gained through
several successive and increasing magnetisations) contains information about the magnetic properties of a
sample. For instance this can be employed to determine the sample's hardness (a term that describes the relative
ease or difficulty with which a sample is magnetised) and to gain an insight into its constituent minerals.
AnhysteriticRemanent Magnetisation (ARM)
The ARM of a sample, sometimes called perfect magnetisation, is similar to the IRM in that it is a measurement
of the magnetisation of a sample after it has been subjected to a known field. However the ARM field is not
applied in the same way as with IRM. Instead of a steady field being applied to the sample, it is magnetised
within an alternating magnetic field, with a steady (d.c.) field applied over the top of ac field, the ac field is
increased to a known maximum and then slowly decreased back to zero. The aim of ARM is to drive the
magnetisation of the sample backward and forward around the origin of its hysteresis loop, while magnetising it
with a small steady field. ARM is generally imparted at a high alternating field measured and then demagnetised
using smaller alternating fields, without the application of the steady field. The demagnetisation of the sample
using known fields builds up an ARM curve, this gives similar information to the IRM curve, however ARM
properties are strongly influenced by grain size, and some minerals ARM properties are very distinctive.
Some Permanent Magnet Materials and Their Applications
Parameters that describe permanent magnet materials are i) the remanence Br [Tesla], (ii) the coercive force of
polarisation HcJ [kA/m], (iii) the coercive force of induction HcB [kA/m] and (iv) the energy
product (BH)max [kJ/m3]. For characterisation of permanent magnet materials the demagnetisation curve
(second quadrant of hysteresis loop) is of importance as shown in the following figure.
Br
Saturation
b
a
HcB
Figure 3-8:
H
HcJ
B=0
The magnetic quality of a permanent magnet is given by the energy product (BH)max andwith the remanence Br,
the theoretical (BH)max is given by
BH max

Br2
4 o
This value is reached when HcB is maximum where
H CB 
Br
o
Classical Permanent Magnet Materials


Aluminium - Nickel - Cobalt Materials (AlNiCo)
Hard Ferrite Materials
Modern Permanent Magnet Materials


Samarium - Cobalt Materials
Neodymium - Iron - Boron Materials
Plastic Bonded Permanent Magnet Materials


Plastic Bonded Hard Ferrite Materials
Plastic Bonded Neodymium - Iron - Boron Materials
(3-27)
Quality Range of Permanent Magnet Materials
indicated years are describing the start of commercial usage
Figure 3-9:
Aluminium - Nickel - Cobalt Materials (AlNiCo)
composition range (% by weight):
Al
Ni
Co
Cu
Nb
Ti
Fe
6-13
13-18
0--42
2-6
0-3
0-9
bal.
Figure 3.10:
structure of anisotropic AlNiCo
The structure is stick – like and there is phase - separation in an Fe - and FeCo - rich phase and a NiAl - rich
phase. The Fe - and FeCo - rich particles form strong magnetic phase while NiAl - rich particles are weak or non
magnetic. The magnetic values are closely linked to the microstructure (depending on the alloy composition and
the heat treatment process). Maximum
aximum application temperature for AlNiCo is between 450 - 500°C.
Manufacturing of AlNiCo
Figure 3.11:
Table 3.xx:
Manufacturing step for AlNiCo magnets
Applications
pplications of AlNiCo magnets
Application
measuring devices
tachometers
fluid level (detection)
couplings (at high temp.
applications)
magnet systems for holding
purposes
Advantages
mechanical stability
small reversible changing of magnetic values with
temperature
high application temperature (ca. 500°C)
easy to magnetise
Disadvantages
contains cobalt which is very expensive
limited design possibilities due to low
coercivity
magnetic stability at reverse fields
Hard Ferrite Materials
MCO3 + 6 Fe2O3 MO · 6 Fe2O3 + CO2 (where M = Ba, Sr, Pb)
Structure
Large ions of oxygen, barium or strontium determine the structure of the crystal lattice (hexagonal dense
packing); the smaller iron ions are located in interstitial sites. the unit cells of the hexaferrite lattice ccontain two
formula units of MFe12O19. The Fe-ions
ions are responsible for the magnetic properties; their moments are aligned
parallel or anti-parallel to the c - axis of the lattice (ferri - magnetic order); an average two of three moments
cancel each other outt and one is externally active. The maximum application temperature is 200°C. H
Hard ferrite
magnets can be produced by (i) isotropic dry pressing without magnetic field (ii) anisotropic dry pressing in a
magnetic field and (iii) anisotropic wet pressing in a magnetic field
Manufacturing of hard ferrite magnets
Figure 3.12:
Manufacturing steps for hard ferrites
Table 3.xxx:
Some properties of hard ferrite magnets
Application
dc motors
brushless dc motors
synchronous motors
water pumps for washing machines, dish
washers and aquarium technique
loudspeakers
couplings
magnet systems for holding applications
Advantage
raw material base (lowest price per energy
unit)
thermal stability at elevated temperatures
easy to magnetise
nearly rigid magnetisation
Disadvantage
low magnetic values (Br, (BH)max)
large temperature coefficient of Js
brittleness (ceramic)
water meters
Rare Earth Magnet Materials (SmCo
SmCo5, Sm2Co17 and Nd2Fe14B)
The crystal lattice of SmCo5 is the basic unit for every rare-earth
rare earth magnet materials. The construction is
hexagonal with the magnetic orientation perpendicular to the basal plane and the basic structure of Sm2Co17 is a
mixture between a rhombohedric and a hexagonal
hexagonal lattice. SmCo magnets have very high coercive fields. On the
other hand Nd2Fe14B with high remanence has similar structure like hexagonal CaCu5 Prototype. Four units
construct one basic unit with a tetragonal crystal structure. The maximum application
on temperature is between
100 - 200°C for Nd2Fe14B, 250°C for SmCo5 and between 300 - 350°C for Sm2Co17.
a
Figure 3.13:
b
(a) structure of high coercive SmCo and (b) structure of NdFeB
Manufacturing of Rare-Earth
Earth Magnets
Figure 3.14:
Manufacturing steps for rare-earth
rare
magnets
Table 3.xxx:
Application of samarium-cobalt
samarium
magnets (SmCo5 and Sm2Co17)
Application
brushless dc motors
Advantage
high thermal stability
(reversible and irreversible)
ABS sensors (reluctance
principle)
in general no corrosion
protection required
couplings
miniaturisation possible
(compared to ferrite and
AlNiCo)
miniaturised sensors and
motors (for high temperature)
applications which require a
low reversible temperature
behaviour
Disadvantage
high raw material price
(Neodymium is very
expensive but less expensive
than samarium)
high magnetic field strength
for magnetisation in saturation
needed (for 2:17 grades)
mechanical manufacture
ability / handling (brittle)
Application of Nd2Fe14B magnets
Table 3.xxx:
Advantage
Disadvantage
brushless dc motors
Application
highest remanence and energy product
ABS sensors (high coercive grades)
good mechanical manufacturing ability /
handling
raw material price compared to SmCo
(no or low cobalt content and
Neodymium is less expensive than Sm)
high reversible temperature coefficients
for Js and HcJ
in many cases corrosion protection
required
for high temperature applications special
grades required (neodymium
dymium is highly
corrosive)
couplings
voice coil motors (e.g. hard disk drive)
loud speaker
insertion devices
Plastic Bonded Magnets
Figure 3.15:
Manufacturing steps of some plastic bonded magnets
Table 3.xxx:
Some properties of plastic bonded Nd2Fe14B magnets
Application
Advantage
Disadvantage
brushless DC motor
shape flexibility
temperature limitations
stepper motors
generators
small tolerances without additional production steps
(e.g. grinding)
production without magnetic fields
air cores
automation of production process possible
coating or complete encapsulation required
for certain application environments.
no oriented grades available for
temperatures up to 100 - 120°C.
sensors
insert technique
bearings
variable magnetisation possible (isotropicmaterial)
Table 3.xxx:
Plastic bonded Hard Ferrite Materials
Application
synchronous motors
stepper motors
shape flexibility
automation possible
Advantage
Disadvantage
high reversible temperature coefficient for J s
lower magnetic values compared to sintered
grades due to combination with binder
couplings
pumps
air cores
insert technique
value added features due to raw material price
high temperature applications with PPS-binder
sensors (position, speed etc.)
orientation and magnetisation during moulding
ANTIFERROMAGNETISM
A behaviour connected with a form of magnetically ordered solid that displays no gross magnetization because
neighbouring dipoles are anti-parallel to each other. The manner in which anti-ferromagnetism comes about can
be illustrated by studying the simplest possible arrangement, one in which there are just two opposed magnetic
sub-lattices. This model is exemplified by the Mn2+ ions in RbMnF3 shown below
Rb
A
B
F
A
B
M
B
A
B
A
The crystal lattice of RbMnF3 allows us to divide the totality of Mn2+ ions into “A” sub-lattice (of which each
member has 6 nearest magnetic neighbours, all from the B sub-lattice) and a “B” sub-lattice having all its nearest
magnetic neighbours from A. If there is a negative exchange interaction between the nearest magnetic
neighbours, the interactions for more remote magnetic neighbours will be zero, and the A and B sub-lattices will
become spontaneously magnetized in opposite directions.
The exchange field can again be used to express the effect of the exchange interactions upon the spin orientation
of a magnetic ion. The exchange field experienced by a Mn2+ ion in the A sub-lattice can be expressed as
H A   M B
(3-28)
By analogy with equation 3-22, the magnetization of the A sub-lattice is
C
H  M B 
MA 
(3-29)
2T
where H is any local field and C/2 is the Curie constant appropriate for ½ of the magnetic ions in the crystal.
Similarly, the B sub-lattice magnetization is
C
H  M A 
2T
An observable susceptibility for an anti-ferromagnet is therefore
M  MB
C
m  A

H
T  TN
MB 
(3-30)
for T>>TN where TN 
SUMMARY
simple
simple
ferromagnet
anti-ferromagnet
ferrimagnet
C
2