3. magnetic properties
(1) introduction
magnetic materials are very important in
electrical engineering
• soft magnetic materials – the materials that
can be easily magnetized and demagnetized
applications: transformer cores, stator and
rotor materials
• hard magnetic materials – cannot be easily
demagnetized (permanent magnets)
applications: loud speakers, telephone
receivers
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(2) magnetic fields and quantities
(a) magnetic fields
ferromagnetic materials such as Fe, Co and Ni
– provide strong magnetic field when
magnetized
magnetism is dipolar up to atomic level
magnetic fields are also produced by currentcarrying conductors
magnetic field of a solenoid is
0.4π n i
H = ────
unit: A/m
l
n : number of turns i : current l : length
1 A/m = 4π × 10-3 Oe (oersteds)
2
(b) magnetic induction
if demagnetized iron bar is placed inside a
solenoid, the magnetic field outside solenoid
increases
the magnetic field due to the bar adds to that
of solenoid - magnetic induction (B)
intensity of magnetization (M) : induced
magnetic moment per unit volume
B = μ0 H + μ0 M = μ0 (H + M)
μ0 : permeability of free space
4π × 10-7 T·m/A
in most cases μ0 M > μ0 H, therefore B ≈ μ0 M
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(c) magnetic permeability and susceptibility
magnetic permeability
B
μ = ──
H
for vacuum μ = μ0 = 4π × 10-7 T·m/A
relative permeability
μ
μ r = ──
μ0
B = μ0 μ r H
relative permeability is measure of induced
magnetic field
magnetic materials that are easily magnetized
have high magnetic permeability
magnetic susceptibility
M
χm = ──
H
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(3) types of magnetism
magnetic fields and forces
are due to intrinsic spin
of electrons
(a) diamagnetism – external magnetic field
unbalances orbiting electrons causing
dipoles that oppose applied field
• very small negative magnetic susceptibility
χm ≈ -10-6
(b) paramagnetism – materials exhibit small
positive magnetic susceptibility
χm ≈ 10-6 to 10-2
• paramagnetic effect disappears when the
applied magnetic field is removed
• produced by alignment of individual dipole
moments of atoms or molecules
• increasing in temperature decreases the
paramagnetic effect
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(c) ferromagnetism – large magnetic fields that
can be retained or eliminated as desired can
be produced
• ferromagnetic elements (such as Fe, Co, Ni)
produce large magnetic fields it is due to
spin of the 3d electrons of adjacent atoms
aligning in parallel directions in microscopic
domains by spontaneous magnetization
• random orientation of domains results in no
net magnetization
• the ratio of atomic spacing to diameter of 3d
orbit must be 1.4 to 2.7 for the parallel
alignment to occur
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(d) magnetic moments of a single unpaired
electron
• each electron spinning about its own axis
has dipole moment μB
eh
μB = ──
4πm
m : electron mass e : electronic charge
• in paired electrons positive and negative
moments cancel
ex. show that the numerical value for a Bohr
magneton is 9.27 × 10-24 A·m2
eh
(1.6 × 10-19 C)(6.63 × 10-34 J·s)
μB = ── = ─────────────
4πm
4π (9.11 × 10-31 kg)
= 9.27 × 10-24 A·m2
C·J·s
(A·s)·(N·m)·s
(A·s)·(kg·m/s2)·m·s
─── = ────── = ──────── = A·m2
kg
kg
kg
ex. calculate the saturation magnetization Ms
and saturation induction Bs for pure Fe.
assuming all magnetic moments due to 4
unpaired 3d electrons are aligned in a
magnetic field
Fe has BCC unit cell with a = 0.287 nm 7
(2 atoms)(4 Bohr magnetons/atom)(9.27 × 10-24 A·m2)
MS = ─────────────────────
(2.87 × 10-10 m)3
= 3.15 × 106 A/m
Bs ≈ μo MS ≈ (4π × 10-7 T·m/A)(3.15 × 106 A/m)
= 3.96 T
(e) antiferromagnetism
Mn, Cr
in presence of magnetic field, magnetic
dipoles align in opposite directions
(f) ferrimagnetism
Fe3O4
ions of ceramics have different magnitudes of
magnetic moments and are aligned in
antiparallel manner creating net magnetic
moments
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(4) effect of temperature on ferromagnetism
• above 0 K, thermal energy causes magnetic
dipoles to deviate from parallel arrangement
• at higher temperature, (Curie temperature)
ferromagnetism is completely lost and
material becomes paramagnetic
• on cooling, ferromagnetic domains reform
• Curie temperature for
Co 1123oC
Ni 358oC
Fe 770oC
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(5) ferromagnetic domains
magnetic dipole moments align themselves in
parallel direction in small-volume regions
called magnetic domains
• when demagnetized by slowly cooling from
above its Curie temperature, domains are
rearranged in random order
no net magnetic moment
• when external magnetic field is applied, the
magnetic domains whose moments are
initially parallel to the applied filed grow
• when domain growth finishes, domain
rotation occurs and domain rotation requires
more energy than domain growth
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(6) types of energies that determine the structure
most stable structure is attained when overall
potential energy is minimum
total magnetic energy of a ferromagnetic
material is the sum of the following energies:
(a) exchange energy – potential energy within
a domain is minimized when all atomic
dipoles are aligned in single direction
the alignment is associated with a positive
exchange energy
(b) magnetostatic energy – potential magnetic
energy produced by its external field
formation of multiple domain reduces
magnetostatic energy
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(c) magnetocrystalline anisotropy energy
magnetization with applied field for a single
crystal varies with crystal orientation
ex. for BCC Fe
saturation magnetization occurs most
easily for the <100> direction
saturation magnetization occurs with
highest applied field for <111> direction
for FCC Ni
the easy directions of magnetization are
<111> and <100> the hard direction
grains at different orientations will reach
saturation magnetization at different field
strength
magnetocrystalline anisotropy energy – the
work done to rotate all domains to reach 12
saturation
(d) domain wall energy
domain wall – the boundary between two
domains whose overall moments are at
different directions
domain changes orientation gradually with a
boundary about 300 atoms wide
large width of domain wall is due to balance
between two forces: exchange force and
magnetocrystalline anisotropy
equilibrium wall width is width at which sum
of two energies are minimum
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(e) magnetostrictive energy
magnetostriction – magnetically induced
reversible elastic strain (Δl/l) the order of 10-6
magnetostrictive energy – the energy due to
mechanical stress created by magnetostriction
it is due to change in bond length caused by
rotation of electron-spin dipole moments
equilibrium domain configuration is reached
when sum of magnetostrictive and domain
wall energies are minimum
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(7) magnetization and demagnetization
hysteresis loop – magnetization loop of a
ferromagnetic material
• magnetization and demagnetization do not
follow same loop
• saturation induction Bs
• once magnetized, remnant induction Br
remains even after demagnetization
• negative field Hc (coercive force) must be
applied to completely demagnetize
• area inside the loop is a measure of work
done in magnetizing and demagnetizing 15
(8) soft magnetic materials
• easily magnetized and demagnetized
• low coercive force and high saturation
induction are desirable properties
• hysteresis energy losses – due to dissipated
energy required to push the domain walls
back and forth
• impurities, crystalline imperfections and
precipitates increase hysteresis energy losses
• eddy current energy losses – induced
electric current causes some stray electric
currents resulting from transient voltage
source of energy loss by electrical resistance
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healing
(a) iron-silicon alloys
• Fe–3 to 4% Si alloys are commonly used
soft magnetic materials
• Si increases electrical resistivity
reduces the eddy-current losses
• Si decreases magnetoanisotropy energy and
increases magnetic permeability
decreases hysteresis core losses
• Si decreases magnetostriction and lower
hysteresis energy losses and transformer
noise
• Si decreases saturation induction and Curie
temperature (disadvantage)
• laminated structure further reduces eddycurrent losses
• decrease in energy loss is also achieved by
using grain oriented silicon sheet
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(b) metallic glasses
• noncrystalline domains
• soft magnetic properties, have combination
of ferromagnetic Fe, Co, Ni with metalloids
B and Si
• used in low-energy core-loss transformers,
magnetic sensors and recording heads
• produced by rapid cooling (106 oC/s) as a
thin film on a rotating copper surface mold
a continuous ribbon of metallic glass (0.001
in. thick and 6 in. wide) is produced
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• strong, hard, flexible and corrosion resistant
• easy movement of domain walls due to
absence of grain boundaries
can be magnetized and demagnetized easily
• very narrow hysteresis loops and low
hysteresis energy loss
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(c) nickel-iron alloys
• higher permeability at lower field because of
low magnetoanisotropy and magnetostrictive
energy
• used in highly sensitive communication
equipments
• 50% Ni alloy – moderate permeability, high
saturation induction
• 79% Ni alloy – high permeability, low
saturation induction
• initial permeability is increased by annealing
in presence of magnetic field
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(9) hard magnetic materials – properties
• high coercive force Hc and remanent
magnetic induction Br
• wide and high hysteresis loops and difficult
to demagnetize
• demagnetizing curves can be used for
comparing the strength of permanent
magnets
1.
2.
3.
4.
5.
6.
7.
8.
9.
Sm(Co, Cu)7.4
SmCo5
bonded SmCo5
alnico 5
Mn-Al-C
alnico 8
Cr-Co-Fe
ferrite
bonded ferrite
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• magnetic potential energy is measured by
maximum energy product (BH)max
(BH)max is the area of largest rectangle that
can be inscribed in the second quadrant of
the hysteresis loop
ex. estimate the maximum energy product
(BH)max for Sm(Co, Cu)7.4
trail 1 (0.8 T × 250 kA/m)
= 200 kJ/m3
trail 2 (0.6 T × 380 kA/m)
= 228 kJ/m3
trail 3 (0.55 T × 420 kA/m)
= 231 kJ/m3
trail 4 (0.5 T × 440 kA/m)
= 220 kJ/m3
the highest value is 231 kJ/m3 22
(a) alnico alloys
alnico : aluminum + nickel + cobalt
• high energy product (BH)max = 40~70 kJ/m3
high remnant induction Br = 0.7~1.5 T
moderate coercivity Hc = 40~160 kA/m
• compositions of the alnico alloys
• produced by casting or powder metallurgy
• structure
single phase BCC at 1250oC
cooling to 750~850oC, α and α’ form
α is rich in Ni and Al, is weakly magnetic
α’ is rich in Fe and Co, is highly magnetic
• if heat treated in magnetic field, α’ becomes
elongated and hence is difficult to rotate –23
high coercivity
(b) rare earth alloys
• very high maximum energy product (BH)max
to 240 kJ/m3 and coercivity to 3200 kA/m
due to unpaired 4f electrons
• SmCo5 single phase magnets
coercivity is based on nucleation and
pinning down of domain walls at surfaces
and grain boundaries
high magnetic strengths with (BH)max in the
range of 130~160 240 kJ/m3
• precipitation-hardened Sm(Co,Cu)2.5 alloy
part of Co substituted by Cu
precipitate produced at low
temperatures and domain walls
are pinned at precipitates
addition of small amount of Fe
and Zr promote the development
of high coercivity
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(c) neodymium-iron-boron magnetic alloys
• produced by powder metallurgy and rapid
solidification melt-spun ribbon process
• highly ferromagnetic Nd2Fe14B grains are
surrounded by nonferromagnetic Nd rich
intergranular phase
• high coercivity and energy product due to
difficulty in reverse nucleating
• used in automotive starting motors
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(d) iron-chromium-cobalt magnetic alloys
• structure and properties analogues to alnico
61% Fe, 28% Cr, 11% Co
• single phase BCC structure forms at elated
temperature (1200oC)
• slow-cooling precipitates of Cr-rich α2
phase forms in a matrix of Fe-rich α1 phase
below 650oC
• particles are elongated by forming to
increase coercivity
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• used in permanent magnets of modern
telephone receivers
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(10) ferrites
• magnetic ceramics made by mixing Fe2O3 with
other oxides and carbonates in powdered form
• domain structure and hysteresis loop similar to
ferromagnets but low magnetic saturation
(a) magnetically soft ferrites
• exhibit ferrimagnetic behavior
• composition: MO·Fe2O3 where M is Fe2+,
Mn2+, Ni2+ or Zn2+
• inverse spinel structure – cubic unit cells
with 8 subcells, each subcell has an FCC
structure, only1/2 of octahedral sites and 1/8
of tetrahedral sites are occupied by metal
ions
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• net magnetic moments in inverse spinel
ferrites
Fe2+ ions
4 unpaired 3d electrons.
Fe3+ ions
5 unpaired 3d electrons.
each unpaired 3d electron has one Bohr
magneton
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ex. calculate the theoretical saturation
magnetization M in A/m and the saturation
induction Bs in tesla for ferrite FeO·Fe2O3
(lattice constant of unit cell is 0.893 nm,
and neglect μH term for Bs)
(4 Bohr magnetons) × 8 = 32 Bohr magnetons
32 Bohr magnetons
-24 A·m2)
M = ————————
×
(9.27
×
10
(8.93 × 10-10 m)3
= 5.0 × 105 A/m
Bs ≈ μoM = (4π × 10-7 T·m/A) × (5.0 × 105 A/m)
= 0.63 T
• useful magnetic properties, good insulators
high electrical resistivity
low eddycurrent losses
• applications: low-signal, memory-core,
audiovisual and recording head
applications
• recording heads are made up of Mn-Zn and
Ni-Zn spinel ferrites
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(b) magnetically hard ferrites
• general formula: MO·Fe2O3 and hexagonal
in crystal structure
• the most important in this group:
barium ferrite (BaO·6Fe2O3) and
strontium ferrite (SrO·6Fe2O3)
• low cost, low density, have a high coercive
force
• high magnetocrystalline anisotropy.
• magnetization takes place by domain wall
nucleation and motion.
• applications: generators, relays, motors,
loudspeakers and door closers
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