Download ESM 1 2015 Coey

A Question
What is Magnetism ?
ESM Cluj 2015
Basic Concepts: Magnetostatics
J. M. D. Coey
School of Physics and CRANN, Trinity College Dublin
Ireland.
1.  Introduction
2.  Magnets
3.  Fields
4.  Forces and energy
5.  Units and dimensions
Comments and corrections please: [email protected]
www.tcd.ie/Physics/Magnetism
This series of three lectures covers basic concepts in magnetism;
Firstly magnetic moment, magnetization and the two magnetic
fields are presented. Internal and external fields are distinguished.
The main characteristics of ferromagnetic materials are briefly
introduced. Magnetic energy and forces are discussed. SI units
are explained, and dimensions are given for magnetic, electrical
and other physical properties.
Then the electronic origin of paramagnetism of non-interacting
electrons is calculated in the localized and delocalized limits. The
multi-electron atom is analysed, and the influence of the local
crystalline environment on its paramagnetism is explained.
Assumed is an elementary knowledge of solid state physics,
electromagnetism and quantum mechanics.
Books
Some useful books include:
•  J. M. D. Coey; Magnetism and Magnetic Magnetic Materials. Cambridge University Press (2010) 614 pp
An up to date, comprehensive general text on magnetism. Indispensable!
•  Stephen Blundell Magnetism in Condensed Matter, Oxford 2001
A good treatment of the basics.
•  D. C. Jilles An Introduction to Magnetism and Magnetic Magnetic Materials, Magnetic Sensors and
Magnetometers, CRC Press 480 pp
:
•  J. D. Jackson Classical Electrodynamics 3rd ed, Wiley, New York 1998
A classic textbook, written in SI units.
•  G. Bertotti, Hysteresis Academic Press, San Diego 2000
A monograph on magnetostatics
•  A. Rosencwaig Ferrohydrodynamics, Dover, Mineola 1997
A good account of magnetic energy and forces
•  L.D. Landau and
The definitive text
E. M. Lifschitz Electrodynamics of Continuous Media Pergammon, Oxford 1989
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1 Introduction
2 Magnetostatics
3 Magnetism of the electron
4 The many-electron atom
5 Ferromagnetism
6 Antiferromagnetism and other magnetic order
7 Micromagnetism
8 Nanoscale magnetism
9 Magnetic resonance
10 Experimental methods
11 Magnetic materials
12 Soft magnets
13 Hard magnets
614 pages. Published March 2010
Available from Amazon.co.uk ~€50
14 Spin electronics and magnetic recording
15 Other topics
Appendices, conversion tables.
www.cambridge.org/9780521816144
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1. Introduction
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Magnets and magnetization
0.03 A m2
3 mm
1.1 A m2
17.2 A m2
10 mm
25 mm
m is the magnetic (dipole) moment of the magnet. It is proportional to volume
m = MV
magnetic moment
volume
magnetization
Magnetization is the intrinsic property of the material;
Magnetic moment is a property of a particular magnet.
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Suppose they are made of
Nd2Fe14B
(M ≈ 1.1 MA m-1)
What are the moments?
Magnetic moment - a vector
Each magnet creates a field
around it. This acts on any
material in the vicinity but
strongly with another magnet.
The magnets attract or repel
depending on their mutual
orientation
↑↑
Weak repulsion
↑↓
Weak attraction
Nd2Fe14B
← ← Strong attraction
← → Strong repulsion
tetragonal
easy axis
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Units
What do the units mean?
m
m – A m2
M – A m-1
1A
Ampère,1821. A current loop or
coil is equivalent to a magnet
m2
∈
m = IA
1A
∈
10,000 A
Permanent magnets
win over electromagnets at small
sizes
area of the loop
m = nIA
number of turns
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C
10,000 turns
m
!
Right-hand corkscrew
Magnetic field H – Oersted’s discovery
I
H
δℓ
r
The relation between electric current and magnetic field
was discovered by Hans-Christian Øersted, 1820.
H
I
C
∫ Hdℓ = I
Ampère’s law
H = I/2πr
If I = 1 A, r = 1 mm
H = 159 A m-1
Right-hand corkscrew
Earth’s field ≈ 40 Am-1
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Magnets and currents – Ampere and Arago’s insight
A magnetic moment is equivalent to a current loop.
Provided the current flows in a plane
m = IA
In general:
m = ½ ∫ r × j(r) d3r
Where j is the current density;
I = j.a
jm = ∇ x M
So m = ½ ∫ r × Idl = I∫ dA = m
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2
long solenoid with n
turns. m = nIA
Space
inversion
Polar
vector
-j
Axial
vector
m
Magnetization curves - Hysteresis loop
M
spontaneous magnetization
remanence
coercivity
virgin curve
initial susceptibility
H
major loop
The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a
magnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains.
A broad loop like this is typical of a hard or permanent magnet.
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This is slightly misleading because the positions of the fundamental and auxiliary fields are reversed in the two equations. The form (2.27) H = B/µ0 − M
is preferable
as it emphasizes the relation of the auxiliary magnetic
Another insofar
Question
field to the fundamental magnetic field and the magnetization of the material.
Maxwell’s equations in a material medium are expressed in terms of all four
fields:
2.4 Magnetic field calculations
What is Magnetostatics ?
∇ · D = ρ,
Electromagnetism (2.46)
In magnetostatics there is no time dependence of B,with
D no
or timeρ. We only
∇ · B = 0,
(2.47)
dependence
have magnetic material and circulating conduction currents in a steady state.
E = −∂ B/∂t,
(2.48)
Conservation of electric charge∇is×expressed
by the equation,
Maxwell’s Equations
∇∇ ×
j + ∂ D/∂t.
· jH==
−∂ρ/∂t,
(2.49)
(2.50)
In
we have
only
magnetic
material
andiscirculating
currents
and
inmagnetostatics,
aρsteady
state ∂ρ/∂t
=charge
0.
The
world ofand
magnetostatics,
we explore
displacement
current.
Here
is the local
electric
density
∂ D/∂t
the which
in
iniswriting
atherefore
steadythe
state.
The
fields
areofproduced
by the magnets
further
in Chapterall7,of
described
by three
simple
equations:
A conductors,
consequence
basic
equations
electromagnetism
in this neat
& the currents
and easy-to-remember form is that the constants µ0 , ϵ 0 and 4π are invisible.
∇ · j = 0 ∇ · B = 0 ∇ × H = j.
But they inevitably crop up elsewhere, for example in the Biot–Savart law (2.5).
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In order
solve problems
in solid-state
to know
theappear
response
The
fieldstonaturally
form two
pairs:
Bphysics
and E we
areneed
a pair,
which
in the
2. Magnetization
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Magnetic Moment and Magnetization
The magnetic moment m is the elementary quantity in solid state magnetism.
Define a local moment density - magnetization – M(r, t) which fluctuates wildly on a subnanometer and a sub-nanosecond scale.
Define a mesoscopic average magnetization
δm = MδV
M = δm/δV
The continuous medium approximation
M can be the spontaneous magnetization Ms within a ferromagnetic domain
A macroscopic average magnetization is the domain average
M (r)
M = ΣiMiVi/ ΣiVi
Ms
atoms
The mesoscopic average magnetization
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Magnetization and current density
The magnetization of a solid is somehow related to a ‘magnetization current density’ Jm that
produces it.
Since the magnetization is created by bound currents,
∫s Jm.dA = 0 over any surface.
Using Stokes theorem ∫ M.dℓ = ∫s(∇ × M).dA and choosing a path of integration outside
the magnetized body, we obtain ∫s M.dA = 0, so we can identify Jm
Jm = ∇ × M
We don’t know the details of the magnetization currents, but we can measure the
mesoscopic average magnetization and the spontaneous magnetization of a sample.
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B and H fields in free space; permeability of free space µ0
When illustrating Ampère’s Law we labelled the magnetic field created by the
current, measured in Am-1 as H. This is the ‘magnetic field strength’
Maxwell’s equations have another field, the ‘magnetic flux density’, labelled B, in the
equation ∇. B = 0. It is a different quantity with different units. Whenever H
interacts with matter, to generate a force, an energy or an emf, a constant µ0, the
‘permeability of free space’ is involved.
In free space, the relation between B and H is simple. They are numerically
proportional to each other
B = µ0H
B-field
Units Tesla, T
Tesla
Permeability
of free space
Units TmA-1
H-field
Units A m-1
µ0 depends on the definition of the Amp.
It is precisely 4π 10-7 T m A-1
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In practice you can
never mix them up.
The differ by
almost a million!
(795775 ≈ 800000)
Field due to electric currents
We need a differential form of Ampère’s Law; The Biot-Savart Law
δℓ
I
δB
B
j
C
Right-hand corkscrew (for the vector product)
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Field due to a small current loop (equivalent to a magnetic moment)
A
BA=4δBsinε
ε
r
sinε=δl/2r
m
δl
m = I(δl)2
B
I
A
So at a general point C, in spherical coordinates:
C
m
msinθ
θ
mcosθ
B
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Field due to a magnetic moment m
The Earth’s magnetic field is roughly
that of a geocentric dipole
tan I = Br /Bθ = 2cotθ
dr/rdθ = 2cotθ
m
Solutions are
r = c sin2θ
Equivalent forms
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Magnetic field due to a moment m; Scaling
m
θ
Why does magnetism lend itself to
miniaturization ?
P
r
HA =
A
2Ma3/4πr3;
H = (m/4πr3){2cosθer- sinθeθ}
Just like the
field of an
electric dipole
What is the average
magnetization of the
Earth ?
•A
If a = 0.01m, r = 2a, M = 1 MA m-1
HA = 2M/16π = 40 kA m-1
Magnet-generated fields just depend
on M. They are scale-independent
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a
m
3. Magnetic Fields
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Magnetic flux density - B
Now we discuss the fundamental field in magnetism.
Magnetic poles, analogous to electric charges, do not exist. This truth is expressed in
Maxwell’s equation
∇.B = 0.
This means that the lines of the B-field always form complete loops; they never
start or finish on magnetic charges, the way the electric E-field lines can start and
finish on +ve and -ve electric charges.
The same can be written in integral form over any closed surface S
∫SB.dA = 0
(Gauss’s law).
The flux of B across a surface is Φ = ∫B.dA. Units are Webers (Wb).
The net flux across any closed surface is zero.
B is known as the flux density; units are Teslas. (T = Wb m-2)
Flux quantum Φ0 = 2.07 1015 Wb
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(Tiny)
The B-field
Sources of B
•  electric currents in conductors
•  moving charges
•  magnetic moments
B = µ0I/2πr
•  time-varying electric fields. (Not in magnetostatics)
In a steady state: Maxwell’s equation
ex
ey
ez
(Stokes theorem;
∂/∂x ∂/∂y ∂/∂z
Bx
BY
I
)
BZ
Field at center of current loop
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Forces between conductors; Definition of the Amp
F = q(E + v x B)
Lorentz expression. Note the Lorentz force does no work on the charge
Gives dimensions of B and E. [T is equivalent to Vsm-1]
If E = 0 the force on a straight wire arrying
a current I in a uniform field B is F = BIℓ
The force between two parallel wires each
carrying one ampere is precisely 2 10-7 N
m-1. (Definition of the Amp)
The field at a distance 1 m from a wire
carrying a current of 1 A is 0.2 µΤ
B = µ0I1/2πr
Force per meter = µ0I1I2/2πr
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The range of magnitude of B
Su
e
th
at
d
ry
el
ta
le
So
Ea
rt
h'
no
s
id
Fi
ne
1E-6
µT
e
ac
Sp
ce
pa
te
rp
la
lla
1E-9
In
In
te
rs
te
H
an
um
1E-12
pT
rS
rt
ea
in
ra
B
H
an
um
H
1E-15
rf
Pe
ac
rm
e
Su a
p n
H e r c ent
yb o M
Pu rid n d ag
u n
Ex lse Ma c t i et
pl M gn n g
os a g et M
ag
iv ne
e t
ne
Fl
t
ux
C
om
N
eu
pr
es
tr
on
si
on
St
ar
M
ag
ne
ta
r
• The tesla is a very large
unitcontinuous laboratory field 45 T
Largest
• Largest continuous field
acheived in a lab was 45 T
(Tallahassee)
1E-3
1
T
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1000
1E6
MT
1E9
TT1E12
1E15
Typical values of B
Electromagnet 1 T
Human brain 1 fT
Magnetar
1012 T
Permanent magnets 0.5 T
Earth 50 µT
Helmholtz coils 0.01 T
Electromagnet 1 T
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Superconducting
magnet 10 T
Sources of uniform magnetic fields
Long solenoid
Helmholtz coils
B =µ0nI
B = (4/5)3/2µ0NI/a
Halbach cylinder
r2
B =µ0M ln(r2/r1)
r1
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The H-field.
Ampère’s law for the field is free space is ∇ x B = µ0( jc + jm) but jm cannot be measured !
Only the conduction current jc is accessible.
We showed on slide 16 that Jm = ∇ × M
Hence ∇ × (B/µ0 – M) = µ0 jc
We can retain Ampère’s law is a usable form provided we define H = B/µ0 – M
Then
∇ x H = µ0 jc
And
B = µ0(H + M)
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The H-field.
The H-field plays a critical role in condensed matter.
The state of a solid reflects the local value of H.
Hysteresis loops are plotted as M(H)
Unlike B, H is not solenoidal. It has sources and sinks in a magnetic material wherever the
magnetization is nonuniform.
∇.H = - ∇.M
The sources of H (magnetic charge, qmag) are distributed
— in the bulk with charge density -∇.M
— at the surface with surface charge density M.en
Coulomb approach to calculate H (in the absence of currents)
Imagine H due to a distribution of magnetic charges qm (units Am) so that
H = qmr/4πr3
[just like electrostatics]
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Potentials for B and H
It is convenient to derive a field from a potential, by taking a spatial derivative. For example E
= -∇ϕe(r) where ϕe(r) is the electric potential. Any constant ϕ0 can be added.
For B, we know from Maxwell’s equations that ∇.B = 0. There is a vector identity
∇. ∇× X ≡ 0. Hence, we can derive B(r) from a vector potential A(r) (units Tm),
B(r) = ∇ × A(r)
The gradient of any scalar f can be added to A (a gauge transformation) This is because of
another vector identity ∇×∇.f ≡ 0.
Generally, H(r) cannot be derived from a potential. It satisfies Maxwell’s equation ∇ × H = jc
+ ∂D/∂t. In a static situation, when there are no conduction currents present, ∇ × H = 0, and
H(r) = - ∇ϕm(r)
In these special conditions, it is possible to derive H(r) from a magnetic scalar potential ϕm
(units A). We can imagine that H is derived from a distribution of magnetic ‘charges’± qm.
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Relation between B and H in a material
The general relation between B, H and M is
B = µ0(H + M)
H
i.e. H = B/µ0 - M
M
B
We call the H-field due to a magnet; — stray field outside the magnet
— demagnetizing field, Hd, inside the magnet
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Boundary conditions
Conditions on the fields
It follows from Gauss’s law
∫SB.dA = 0
that the perpendicular component of B is
continuous. (B1- B2).en= 0
It follows from from Ampère’s law
∫ H.dl = I = 0
that the parallel component of H is continuous.
since there are no conduction currents on the
surface. (H1- H2) x en= 0
Conditions on the potentials
Since ∫SB.dA = ∫ A.dl (Stoke’s theorem)
(A1- A2) x en= 0
The scalar potential is continuous ϕm1 = ϕm2
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Boundary conditions in linear, isotropic homogeneous media
In LIH media,
Hence
B = µ 0 µ rH
B1en = B2en
H1en = µr2/µr1H2en
So field lies ~ perpendicular to the surface of soft iron but parallel to the surface of a
superconductor.
Diamagnets produce weakly repulsive images.
Paramagnets produce weakly attractive images.
Soft ferromagnetic mirror
Superconducting mirror
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Field calculations – Volume integration
Integrate over volume distribution of M
Sum over fields produced by each magnetic
dipole element Md3r.
Using
Gives
(Last term takes care of divergences at the origin)
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Field calculations – Ampèrian approach
Consider bulk and surface current distributions
jm = ∇ x M
and
jms = M x en
Biot-Savart law gives
For uniform M, the Bulk term is zero since ∇ x M = 0
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Field calculations – Coulombian approach
Consider bulk and surface magnetic charge distributions
ρm = -∇.M
and
ρms = M.en
H field of a small charged volume element V is
δH = (ρmr/4πr3) δV
So
For a uniform magnetic distribution the first term is zero.
∇.M = 0
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Hysteresis loop; permanent magnet (hard ferromagnet)
M
spontaneous magnetization
remanence
coercivity
virgin curve
initial susceptibility
major loop
A broad M(H) loop
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H
Hysteresis loop; temporary magnet (soft ferromagnet)
Ms
Slope here is the initial
susceptibility χi > 1
M
H
’
Applied field
Susceptibility is
defined as χ = M/H
A narrow M(H) loop
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Paramagnets and diamagnets; antiferromagnets.
Only a few elements and alloys are ferromagnetic. (See the magnetic periodic table).
The atomic moments in a ferromagnet order spontaneosly parallel to eachother.
Most have no spontaneous magnetization, and they show only a very weak response
to a magnetic field.
M
paramagnet
H
Here |χ| << 1
χ is
10-4
-
diamagnet
10-6
Ordered T < Tc
A few elements and oxides are
antiferromagnetic. The atomic
moments order spontaneously
antiparallel to eachother.
Disordered T > Tc
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Magnetic Periodic Table
1H
1.00
66Dy
Atomic Number
3 Li
4 Be
6.94
1 + 2s0
9.01
2 + 2s0
11Na
12Mg
19K
20Ca
22.99
1 + 3s0
38.21
1 + 4s0
Typical ionic change
Antiferromagnetic TN(K)
40.08
2 + 4s0
Sr
21Sc
44.96
3 + 3d0
39
Y
22Ti
23V
47.88
4 + 3d0
40
Zr
50.94
3 + 3d2
41
Nb
24Cr
52.00
3 + 3d3
312
42
88.91
2 + 4d0
91.22
4 + 4d0
92.91
5 + 4d0
95.94
5 + 4d1
55Cs
56Ba
57La
72Hf
73Ta
74W
87Fr
88Ra
89Ac
58Ce
59Pr
226.0
2 + 7s0
138.9
3 + 4f0
178.5
4 + 5d0
227.0
3 + 5f0
180.9
5 + 5d0
140.1
4 + 4f0
13
90Th
232.0
4 + 5f0
25Mn 26Fe
55.85
2 + 3d5
96
Mo 43 Tc
87.62
2 + 5s0
223
5B
6C
7N
8O
13Al
14Si
15P
183.8
6 + 5d0
140.9
3 + 4f2
91Pa
231.0
5 + 5f0
97.9
55.85
3 + 3d5
1043
44
Ru
27Co
58.93
2 + 3d7
1390
45
Rh
28Ni
58.69
2 + 3d8
629
46
Pd
29Cu
31Ga
32Ge
33As
63.55
2 + 3d9
47
Ag
12.01
14.01
9F
10Ne
16S
17Cl
18Ar
34Se
35Br
36Kr
I
54Xe
16.00
19.00
30Zn
65.39
2 + 3d10
48
Cd
69.72
3 + 3d10
49
In
28.09
72.61
50
Sn
30.97
74.92
51
Sb
32.07
78.96
52
Te
35.45
79.90
53
39.95
83.80
101.1
3 + 4d5
102.4
3 + 4d6
106.4
2 + 4d8
107.9
1 + 4d10
112.4
2 + 4d10
114.8
3 + 4d10
118.7
4 + 4d10
121.8
127.6
126.9
75Re
76Os
77Ir
78Pt
79Au
80Hg
81Tl
82Pb
83Bi
84Po
85At
60Nd
61Pm 62Sm 63
Eu
64Gd 65Tb
66Dy
67Ho
68Er
69Tm 70Yb
95Am 96Cm 97Bk
98Cf
99Es 100Fm 101Md 102No 103Lr
186.2
4 + 5d3
144.2
3 + 4f3
19
92U
238.0
4 + 5f2
190.2
3 + 5d5
192.2
4 + 5d5
145
150.4
3 + 4f5
105
93Np
94Pu
238.0
5 + 5f2
244
195.1
2 + 5d8
152.0
2 + 4f7
90
197.0
1 + 5d10
157.3
3 + 4f7
292
243
247
200.6
2 + 5d10
204.4
3 + 5d10
158.9
162.5
3 + 4f8
3 + 4f9
229 221 179 85
247
251
207.2
4 + 5d10
164.9
3 + 4f10
132 20
252
209.0
167.3
3 + 4f11
85 20
257
209
168.9
3 + 4f12
56
210
173.0
3 + 4f13
258
Nonmetal
Diamagnet
Ferromagnet TC > 290K
Metal
Paramagnet
Antiferromagnet with TN > 290K
Magnetic atom
Antiferromagnet/Ferromagnet with TN/TC < 290 K
Radioactive
20.18
35
26.98
3 + 2p6
85.47
1 + 5s0
137.3
2 + 6s0
4.00
10.81
Ferromagnetic TC(K)
24.21
2 + 3s0
37Rb 38
132.9
1 + 6s0
Atomic symbol
Atomic weight
162.5
3 + 4f9
179 85
2 He
BOLD
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259
83.80
86Rn
222
71Lu
175.0
3 + 4f14
260
Susceptibilities of the elements
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Magnetic fields - Internal and applied fields
In ALL these materials, the H-field acting inside the material is not the one you apply.
These are not the same. If they were, any applied field would instantly saturate the
magnetization of a ferromagnet when χ > 1.
M
Consider a thin film of iron.
iron
Ms
Field applied
|| to film
Field applied
⊥ to film
Hʹ′
Ms
substrate
H = H’ + Hd
Internal field
Demagnetizing field
External field
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Ms = 1.72 MAm-1
Demagnetizing field in a material - Hd
The demagnetizing field depends on the shape of the sample and the direction of
magnetization.
For simple uniformly-magnetized shapes (ellipsoids of revolution) the demagnetizing
field is related to the magnetization by a proportionality factor N known as the
demagnetizing factor. The value of N can never exceed 1, nor can it be less than 0.
Hd = - N M
More generally, this is a tensor relation. N is then a 3 x 3 matrix, with trace 1. That is
Nx+Ny+Nz=1
Note that the internal field H is always less than the applied field H’ since
H = H’ - N M
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Demagnetizing factor N for special shapes.
Shapes N = 0
N = 1/3
N = 1/2
N =1
H
’
H
’
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H
’
The shape barrier
N < 0.1
Shen Kwa 1060
Daniel Bernouilli
1743
S
N
Gowind Knight 1760
T. Mishima
1931
N = 0.5
New icon for permanent magnets! ⇒
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The shape barrier ovecome !
Hd = - N M
working
point
Ms
M (Am-1)
Mr
Hd
Hc
Hd
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H (Am-1)
Demagnetizing factors N I for a general ellipsoid.
Calculate from analytial
formulae
Nx+Ny+Nz=1
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Other shapes cannot be unoformly magnetized..
When measuring the magnetization of a sample H is always
taken as the independent variable, M = M(H).
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Paramagnetic and ferromagnetic responses
Susceptibility of linear, isotropic and homogeneous (LIH) materials
M = χ’H’
χ’ is external susceptibility (no units)
It follows that from H = H’ + Hd that
1/χ = 1/χ’ - N
Typical paramagnets and diamagnets:
χ ≈ χ’
(10-5 to 10-3 )
Paramagnets close to the Curie point and ferromagnets:
χ diverges as T → TC but χ’ never exceeds 1/N.
χ >> χ’
M
M
M
M
M
H’
M
/3
Mss/3
H’
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H'
H
Demagnetizing correction to M(H) and B(H) loops.
N=0
N=½
The B(H) loop is deduced from M(H) using B = µ0(H + M).
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N=1
Permeability
In LIH media
B =µ H
Relative permeability is defined as
B = µ0(H + M)
gives
defines the permeability µ
µr = µ / µ0
µr = 1 + χ
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Units: TA-1m
4. Energy and forces
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Energy of ferromagnetic bodies
Torque and potential energy of a dipole in a field, assumed to be constant.
Force
Γ=mxB
ε = -m.B
Γ = mB sinθ.
ε = -mB cosθ.
In a non-uniform field,
f = -∇εm
B θ m
f = m.∇B
•  Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic
microstructure (domains).
•  ½µ0H2 is the energy density associated with a magnetic field H
M ≈ 1 MA m-1, µ0Hd ≈ 1 T, hence µ0HdM ≈ 106 J m-3
•  Products B.H, B.M, µ0H2, µ0M2 are all energies per unit volume.
•  Magnetic forces do no work on moving charges f = q(v x B) [Lorentz force]
•  No potential energy associated with the magnetic force.
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Reciprocity theorem
The interaction of a pair of dipoles, εp, can be considered as the energy
of m1 in the field B21 created by m2 at r1 or vice versa.
εp = -m1.B21 = -m2.B12
So
εp = -(1/2)(m1.B21+ m2.B12)
Extending to magnetization distributions:
ε = -µ0 ∫ M1.H2 d3r = -µ0 ∫ M2.H1 d3r
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Self Energy
The interaction of the body with the field it creates. Hd.
Consider the energy to bring a small moment δm into position within the
magnetized body
Hloc=Hd+(1/3)M
δε = - µ0 δm.Hloc
Integration over the whole sample gives
The magnetostatic self energy is defined as
Or equivalently, using B = µ0(H + M) and ∫ B.Hdd3r=0
For a uniformly magnetized ellipsoid
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Energy associated with a field
General expression for the energy associated with a magnetic field distribution
Aim to maximize energy associated with the field created around the magnet,
from previous slide:
Can rewrite as:
where we want to maximize the integral on the left.
Energy product: twice the energy stored in the stray field of the magnet
-µ0 ∫i B.Hd d3r
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Work done by an external field
Elemental work δw to produce a flux change δΦ is I δΦ
Ampere: ∫H.dl = I So δw = ∫ δΦ H.dl
So in general:
H = H´+ Hd
δw = ∫ δB.H.d3r
B = µ0(H + M)
Subtract the term associated with the H-field in empty space, to give the
work done on the body by the external field;
gives
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Thermodynamics
First law:
dU = HxdX + dQ
(U ,Q, F, G are in units of Jm-3)
M
dQ = TdS
Four thermodynamic potentials;
ΔG
U(X,S)
E(HX,S)
− ΔF
F(X,T) = U - TS
dF = HdX - SdT
G(HX,T) = F- HXX dG = -XdH - SdT
Magnetic work is HδB or µ0H’δM
dF = µ0H’dM - SdT
dG = -µ0MdH’ - SdT
H’
S = -(∂G/∂T)H’
µ0 M = -(∂G/∂H’)T’
Maxwell relations
(∂S/∂H’)T’ = - µ0(∂M/∂T)H’ etc.
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Magnetostatic forces
Force density on a magnetized body at constant temperature
Fm= - ∇ G
Kelvin force
General expression, for when M is dependent on H is
V =1/d
d is the density
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Anisotropy - shape
The energy of the system increases when the
magnetization deviates by an angle θ from its easy axis.
θ
E = K sin2 θ
The anisotropy may be due to shape. The energy of the
magnet in the demagnetizing field is -(1/2)µ0mHd
For shape anisotropy Ksh = (1/4)µ0M2(1 - 3N )
For example, if M = 1 MA m-1, N = 1, Ksh = 630 kJ m-3
e.g. if a thin film is to have its magnetization perpendicular to the plane, there
must be a stronger intrinsic magnetocrystalline anisotropy K1.
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Polarization and anisotropy field
Engineers often quote the polarization of a magnetic material, rather than its
magnetization. The relation is simple; J = µ0M.
For example, the polarization of iron is 2.15 T; Its magnetization is 1.71 MA m-1
The defining relation is then:
B = µ0H + J
The anisotropy of whatever origin may be represented by an effective magnetic
field, the anisotropy field Ha
Ha = 2K/µ0Ms
The coercivity can never exceed the anisotropy field. In practice it is rarely
possible to obtain coercivity greater than about 25 % of Ha. Coercivity depends on
the microsctructure, and the ability to impede fprmation of reversed domains
Sometimes anisotropy field is quoted in Tesla. Ba = µ0Ha.
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Some expressions involving B
F = q(E + v × B)
F = B lℓ
Force on a charged particle q
Force on current-carrying wire
E = - dΦ/dt
Faraday’s law of electromagnetic induction
E = -m.B Energy of a magnetic moment
F = ∇ m.B
Force on a magnetic moment
Γ=mxB
Torque on a magnetic moment
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5. Units
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A note on units:
Magnetism is an experimental science, and it is important to be able to
calculate numerical values of the physical quantities involved. There is a
strong case to use SI consistently
Ø  SI units relate to the practical units of electricity measured on the
multimeter and the oscilloscope
Ø  It is possible to check the dimensions of any expression by
inspection.
Ø They are almost universally used in teaching
Ø  Units of B, H, Φ or dΦ/dt have been introduced.
BUT
Most literature still uses cgs units,You need to understand them too.
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SI / cgs conversions:
SI units
cgs units
B = μ0(H + M)
B = H + 4πM
m
A m2
emu
M
A m-1
(10-3 emu cc-1)
emu cc-1
(1 k A m-1)
σ
A m2 kg-1
(1 emu g-1)
emu g-1
(1 A m2 kg-1)
H
A m-1 (4π/1000 ≈ 0.0125 Oe)
Oersted
B
Tesla
(10000 G)
Gauss
(10-4 T)
Φ
Weber (Tm2)
(108 Mw)
Maxwell (G cm2)
(10-8 Wb)
dΦ/dt V
(108 Mw s-1)
Mw s-1
(10 nV)
χ
(4π cgs)
-
(1/4π SI)
-
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(1000/4π ≈ 80 A m-1)
B3.1 Dimensions
Mechanical
Quantity
Symbol
Unit
m
l
t
i
θ
Area
Volume
Velocity
Acceleration
Density
Energy
Momentum
Angular momentum
Moment of inertia
Force
Force density
Power
Pressure
Stress
Elastic modulus
Frequency
Diffusion coefficient
Viscosity (dynamic)
Viscosity
Planck’s constant
A
V
v
a
d
ε
p
L
I
f
F
P
P
σ
K
f
D
η
ν
!
m2
m3
m s−1
m s−2
kg m−3
J
kg m s−1
kg m2 s−1
kg m2
N
N m−3
W
Pa
N m−2
N m−2
s−1
m2 s−1
N s m−2
m2 s−1
Js
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
1
2
3
1
1
−3
2
1
2
2
1
−2
2
−1
−1
−1
0
2
−1
2
2
0
0
−1
−2
0
−2
−1
−1
0
−2
−2
−3
−2
−2
−2
−1
−1
−1
−1
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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Thermal
Electrical
Quantity
Symbol
Unit
m
l
t
i
θ
Current
Current density
Charge
Potential
Electromotive force
Capacitance
Resistance
Resistivity
Conductivity
Dipole moment
Electric polarization
Electric field
Electric displacement
Electric flux
Permittivity
Thermopower
Mobility
I
j
q
V
E
C
R
ϱ
σ
p
P
E
D
%
ε
S
µ
A
A m−2
C
V
V
F
"
"m
S m−1
Cm
C m−2
V m−1
C m−2
C
F m−1
V K−1
m2 V−1 s−1
0
0
0
1
1
−1
1
1
−1
0
0
1
0
0
−1
1
−1
0
−2
0
2
2
−2
2
3
−3
1
−2
1
−2
0
−3
2
0
0
0
1
−3
−3
4
−3
−3
3
1
1
−3
1
1
4
−3
2
1
1
1
−1
−1
2
−2
−2
2
1
1
−1
1
1
2
−1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
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Magnetic
Mobility
µ
m V
s
−1
0
2
1
0
Magnetic
Quantity
Symbol
Unit
m
l
t
i
θ
Magnetic moment
Magnetization
Specific moment
Magnetic field strength
Magnetic flux
Magnetic flux density
Inductance
Susceptibility (M/H)
Permeability (B/H)
Magnetic polarization
Magnetomotive force
Magnetic ‘charge’
Energy product
Anisotropy energy
Exchange stiffness
Hall coefficient
Scalar potential
Vector potential
Permeance
Reluctance
m
M
σ
H
'
B
L
χ
µ
J
F
qm
(BH )
K
A
RH
ϕ
A
Pm
Rm
A m2
A m−1
A m2 kg−1
A m−1
Wb
T
H
0
0
−1
0
1
1
1
0
1
1
0
0
1
1
1
0
0
1
1
−1
2
−1
2
−1
2
0
2
0
1
0
0
1
−1
−1
1
3
0
1
2
−2
0
0
0
0
−2
−2
−2
0
−2
−2
0
0
−2
−2
−2
−1
0
−2
−2
2
1
1
1
1
−1
−1
−2
0
−2
−1
1
1
0
0
0
−1
1
−1
−2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
H m−1
T
A
Am
J m−3
J m−3
J m−1
m3 C−1
A
Tm
T m2 A−1
A T−1 m−2
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Thermal
594
Quantity
Symbol
Unit
Enthalpy
Entropy
Specific heat
Heat capacity
Thermal conductivity
Sommerfeld coefficient
Boltzmann’s constant
H
J
S
J K−1
C
J K−1 kg−1
c
J K−1
κ
W m−1 K−1
Appendices
γ
J mol−1 K−1
J K−1
kB B3.2 Examples
(1) Kinetic energy of a body: ε = 12 mv 2
[ε] = [1, 2, −2, 0, 0]
m
l
t
i
θ
1
1
0
1
1
1
1
2
2
2
2
1
2
2
−2
−2
−2
−2
−3
−2
−2
0
0
0
0
0
0
0
0
−1
−1
−1
−1
−1
−1
[m] = [1, 0, 0, 0, 0]
2[0, −1, −1, 0, 0]
[v 2 ] =
[1, −2, −2, 0, 0]
(2) Lorentz force on a moving charge; f = qv × B
[f ] = [1, 1, −2, 0, 0]
[q] = [0, 0, 1, 1, 0]
[v] = [0, 1, −1, 0, 0]
[1, 0, −2, −1, 0]
[B] =
[1, 1, −2, 0, 0]
√
(3) Domain wall energy γ w = AK (γ w is an energy
per
unit area)
√
−1
[ AK] = 1/2[AK]
[γ w ] = [εA ]
√
= [1, 2, −2, 0, 0]
[ A] = 12 [1, 1, −2, 0, 0]
√
[1, −1, −2, 0, 0]
−[
[ K] = 12
- [0,1,2,1,0,−2,
0, 0 0,
] 0]
[1, 0, −2, 0, 0]
= [1, 0, −2, 0, 0]
(4) Magnetohydrodynamic force on a moving conductor F = σ v × B × B
(F is a force per unit volume) ESM Cluj 2015
[σ ] = [−1, −3, 3, 2, 0]
[F ] = [F V −1 ]
√
[ K] =
−[ 1, 1, −2, 0, 0]
(4)
(5)
(6)
(7)
(8)
1
2
[1, −1, −2, 0, 0]
[1, 0, −2, 0, 0]
= [1, 0, −2, 0, 0]
Magnetohydrodynamic force on a moving conductor F = σ v × B × B
( F is a force per unit volume)
[σ ] = [−1, −3, 3, 2, 0]
[F ] = [F V −1 ]
= [1, 1, −2, 0, 0]
[v] = [0, 1, −1, 0, 0]
2[1, 0, −2, −1, 0]
[0, 3, 0, 0, 0]
[B 2 ] =
−
[1, −2, −2, 0, 0]
[1, −2, −2, 0, 0]
Flux density in a solid B = µ0 ( H + M) (note that quantities added or
subtracted in a bracket must have the same dimensions)
[B] = [1, 0, −2, −1, 0]
[µ0 ] = [1, 1, −2, −2, 0]
[0, −1, 0, 1, 0]
[M], [H ] =
[1, 0, −2, −1, 0]
Maxwell’s equation ∇ × H = j + d D/dt .
[j ] = [0, −2, 0, 1, 0] [d D/dt ] = [Dt −1 ]
[∇ × H] = [H r −1 ]
= [0, −1, 0, 1, 0]
= [0, −2, 1, 1, 0]
−[ 0, 1, 0, 0, 0]
−[0, 0, 1, 0, 0]
= [0, −2, 0, 1, 0]
= [0, −2, 0, 1, 0]
Ohm’s Law V = I R
= [1, 2, −3, −1, 0]
[0, 0, 0, 1, 0]
+ [1, 2, −3, −2, 0]
= [1, 2, −3, −1, 0]
Faraday’s Law E = −∂%/∂t
[1, 2, −2, −1, 0]
= [1, 2, −3, −1, 0]
−[0, 0, 1, 0, 0]
= [1, 2, −3, −1, 0]
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