27-445 Structure, Properties and Performance Relationships in
Magnetic Materials - Spring 2023
Prof. M. E. McHenry, REH 243; (Office Hours by arrangement)
Secretary: Mary Grace Antkowski, REH 138; x8-7240 ma5@andrew.cmu.edu
Graders: TBD
TR 12:30-01:50
Porter Hall 125D
Text: J. M. D. Coey,
Magnetism and Magnetic Materials.
Cambridge University Press (2010).
Course Notes: Course notes will be posted on the Canvas System.
Course notes are at a graduate level, Powerpoint Lectures Notes and
assignments will emphasize topics at an UG level. A pdf
version of lectures will be posted on Canvas.
Grading:
25% HW
25% Midterm Exam
50% Project and Presentation
January 17, 2023
January 19, 2023
January 24, 2023
January 26, 2023
January 31, 2023
February 02, 2023
February 07, 2023
February 09, 2023
February 14, 2023
February 16, 2023
February 21, 2023
February 23, 2023
February 28, 2023
March 02, 2023
Units, Fields, Overview
Magnetostatics
QM – Local Dipoles
Dipoles – Energy Bands
Paramagnetism
Collective Magnetism
Exchange
Thermo - Phase Diagrams
Magnetocaloric/Landau theory I
Magnetic Anisotropy
Magnetostriction
Magnetic Domains I
Landau/Magnetocaloric - II
Midterm Exam*
March 06 – 10
Spring Break
01
02
03
04
05
06
07
08
09
10
11
12
13
14
HW#1
HW#2 HW#1due
HW#3 HW#2due
HW#4 HW#3due
HW#5 HW#4due
HW#5 due
March 14, 2023
March 16, 2023
March 21, 2023
March 23, 2023
March 28, 2023
March 30, 2023
April 04, 2023
April 06, 2023
April 11, 2023
Domains –II Skyrmions
Ferrites - I
Ferrites - II, Microwave.
Hard Magnets – I
Hard Magnets – II
Ferromagnetic Shape Memory
Magnetotransport
Power Losses & EMI
Recording Media
April 13, 2023
Break No Classes - Carnival
April 18, 2023
April 20, 2023
April 25, 2023
April 27, 2023
Amorphous Magnets
Ferrofluids/Biomagnetic Apps
Special Topic
Final Presentations*
(TBD)
15
16
17
18
19
20
21
22
23
24
25
26
27-445 Lecture One
Coey – Chapter 1,2
Maxwell’s Relationships
Gauss’s Law for an Electric Field
Ampere’s Law
Ampere - Maxwell Law
Faraday’s Law
Gauss’s Law: Magnetic Fields
Net flux of B across a surface = 0
Ampere - Maxwell without a current term and Faraday’s Law:
Equations transform into one another E à B and B à E implying
a symmetry relation or duality between the fields.
The two Maxwell’s equations:
state the divergences of magnetic and electric field are zero or the
enclosed charge, respectively. This is a mathematical way of saying
that there are no magnetic monopoles!
Magnetic Dipole Moment
moment = IA (Am2)
moment
= magnetization
volume
M=
åµ
atoms
V
atom
Magnetic Dipole Moment
Spin and orbital moments add according to quantum
mechanic principles.
M=
åµ
atoms
V
atom
Divergences: No magnetic monopoles à Time reversal symmetry
A magnetic field, B, is an axial as opposed to a polar vector. Axial
vectors form from the cross product of two polar vectors. The axial
vector direction is arbitrary: usually chosen by the right hand rule.
Axial vectors: rotation, angular velocity, moment of force, magnetic
field strength. Polar Vectors: displacement, velocity, acceleration,
force, etc.
Changing the direction of time (time reversal) changes the direction
of the motion of the electron and hence direction of a dipole moment.
Magnetic Fields: may be due to electric
currents or also to moments
Falls off as the
square of the
distance
JA-1m-2 = JA-2m-1 A m-1
Field created by a current
Maxwell’s Relationships
Gauss’ Law:
Lorentz Force
The force on a charge is perpedicular to the velocity and magnetic
field. For E = 0:
For B = 0:
Units and Conversions - NIST
Gaussian (cgs)
SI (mks)
Magnetic Fields
Constitutive Relationships
Magnetic Induction. B Magnetic Field Intensity, H
In vacuum: The magnetic induction, B, is related to the magnetic field
through the relationship:
B = µ0H (mksa, SI) or B = H (cgs)
µo = permeability of the vacuum = 4π x 10-7 H/m (SI)
(free space) = 1 in CGS units.
i.e.
Induction and field are equivalent in vacuo.
In a medium:The magnetic induction can be enhanced or reduced by the
material's magnetization (dipole moment per unit volume)
∑ Dipole moments
M = Magnetization = atoms
Volume
= Net dipole moment per unit volume
Example: for an atomic density, Na and an average atomic dipole
moment < µatom > ; M = Na< µatom >
B = µ0(H+M) (mksa, SI) or B = H+4πM (cgs)
Susceptibility, c is a second rank tensor:
M = cH
Linear response
c = Susceptibility = dM /dH = M /H (linear)
B = µ0(H+ cmH) (mksa, SI)
or B = (1+4π cm)H (cgs)
µr = relative permeability
= µ0(1+ cm) (SI)
= 1+4π cm (cgs)
Figures of Merit for Soft Materials:
Induction, Permeability
Lines of Magnetic Induction near a paramagnetic sphere ( > 0)
B has a larger density inside the paramagnet B = (H + M)
Field Generation and Geometries
Fields result from currents:
Ampere’s Law:
4π
∇×H=
J (cgs) = µ 0 J (SI)
c
Stoke’s Theorem:
4π
∫ ( ∇ × H) ⋅dA = ∫ H ⋅ dl =
∫ J ⋅ dA (cgs)
c S
S
4π
2πrH =
I (cgs)
c
2I
1 2I
H = (cgs) =
(SI)
2
cr
4πε 0 c r
I
or in Engineering units : H = 0.2
r
(Oe)
= 0.2 (A/cm)
Fields result from currents:
Ampere Definition:
Stoke’s Theorem:
Magnetic Field Intensity
(a) Fields result from currents: 3 depictions of a surface current
rotating
charge
A
wire wound
sphere
uniformly
magnetized sample
I
(b) Solenoid:
4π
4π A
nI (
)Oersteds
10
10 cm
n = turns/length
I = current
H=
L
V
Magnetic Field Intensity
(c) Toroid:
(d) Electromagnet :
Magnetic Circuit
4π
4π A
4π
H=
nI (
)Oersteds; H =
nI (cgs)
10
10 cm
c
N
n = turns/length =
2πR
I = current
Ha =
0.4πNI
;
L
(
L = Lg + Lm ; n =
N
L
)
H a L g + L m = 0.4πNI = H m L m + Hg L g
for L g << L m : H g ≈ Bm = H m + ( Bm − H m )
(
)
H m L g + L m = H m L m + [ H m + ( Bm − H m )]L g
Hm = Ha −
Lg
Lg + Lm
(Bm − H m ) Demag
Magnetic Field Intensity
(e) Field Enhancement: Pole Pieces
H = 4πM(1 - cosθ )
θ
θ
typical : 4πM = 1.5 T, 4.0 inch diameter; 2.0 inch gap
r2 r1
# r2 &
r
H = 4πM(1 - cosθ + cosθsin θ ln % ( ); 2 ≈ 2
r1
$ r1 '
2
∂B
1 ∂B
(SI) = −
(cgs)
∂t
c ∂t
1 ( ∂ B+
1 ∂Φ
∫∫ (∇ × E) ⋅ dA = ∫ E ⋅dl = E (2πr ) = − ∫∫ * - ⋅ dA = −
c ) ∂t ,
c ∂t
(f) Pick-up Coils (Faraday’s law) ∇ × E = −
N ∂Φ
-8 ∂Φ
(cgs) = -10 N
(Eng)
c ∂t
∂t
2+
(
-8 * G - cm Eng : Volts = -10 N*
s
)
,
NE (2πr) = V = −
V
B
Units
L
1. SI:Parallel Wires
F12 = Il B2
µ 0I 2
µ 0 I1I 2
B2 =
F12 =
2πr
2πr
Units of H:
Units of B:
Units of M:
SI: A/m
SI: Wb/m2
SI: Wb/m2
X
I2
X
r
1 Amp of Current in two infinite || conductors 1 m
apart results in a force of exactly 2 x 10-7 N/m.
2.
I1
cgs: Oe
1 Oe = 79.6 A/m
cgs: G
cgs: 1/4π x 104 G
At P: B = 1 Wb/m2
1m
therfore 4π Wb/pole
Pole
F12
F21
Dipole Moments – EM/QM
QM
Dipole-Dipole Interactions
Coulomb’s Law:
Diamagnetism
Diamagnetism
Types of Magnetism
Mean Field Theory: Weiss Molecular Field
Heisenberg Exchange
Types of Magnetism
Types of Magnetism
H
Paramagnet
Antiferromagnet
Ferromagnet
Ferrimagnet
Helimagnet
The Demagnetizing Field
The Demagnetizing Field
Demagnetization and Domain structure
Consider a monodomain bar magnet with M = Ms
Hd = Self- (demagnetization) field
µ = Ms (Volume)
Over much of the volume Hd opposes the
magnetization (antiparallel)
Self, Demagnetization Energy
(
)
E d (or E ms ) = òòò M × H d dV
How can this energy cost be reduced? Domains
òòò (M × H
d
)dV ~ 0
Domains
Demagnetization Field
Domain Walls: N atoms spaced a
apart, rotation of π
z in a thickness, t = Na
z
x
JS2p 2
s TOT = s ex + s K =
2 + K1Na
Na
y
Magnetic Response & Hysteresis
Reversible vs. Irreversible
Permanent Magnets: Monodomain
If Hd > HC then:
demag --> multidomain:
HC is determined by rotational
energy barriers to reversal of the
magnetization.
Length Scales
Permanent Magnets: Prevention of Demagnetization
Geometry:
Horseshoe magnet
Toroid with gap (electromagnet)
Chemistry, Microstructure (O’Handley)
Lorentz Force - II