Magnetic Materials
•
The inductor
Φ B = LI (Q = CV)
1 ∂B
(CGS)
∇× E = −
c ∂t
1 ∂
1 ∂Φ B
EdS
BdS
∇
×
=
−
=
−
∫∫
c ∂t ∫ ∫
c ∂t
Φ B ≡ magnetic flux density
(
)
∫ ∫ ∇ × EdS = ∫ E ⋅ dl (Green's Theorem)
V = ∫ E ⋅ dl = −
1 ∂Φ B
(explicit Faraday's Law)
c ∂t
∂Φ B
∂I
=L
∂t
∂t
∂NΦ B
∂I
= −L
VEMF = −
∂t
∂t
∂I
∂V
(recall I = C
for the capacitor)
V =L
∂t
∂t
∂I
Power = VI = LI
∂t
1
1
Energy = ∫ Power ⋅ dt = ∫ LIdI = LI 2 = NΦ B I
2
2
1
⎛
2⎞
⎜ capacitor CV ⎟
2
⎝
⎠
1
©1999 E.A. Fitzgerald
The Inductor
∇× B =
4π
1 ∂E
J+
c
c ∂t
∫ ∫ ∇ × BdS = ∫ B ⋅ dl =
4π
4π
J
⋅
dS
=
I
∫
∫
c
c
4π
In
c
N = n ⋅ length = nl
B=
L=
Nφ B N (BA) 4π 2
=
=
n lA
I
I
c
Insert magnetic material
Magnetic dipoles in material can line-up in magnetic field
B = H + 4πχH = H + 4πM
∂M
= χ μ = 1+ 4πχ
∂H
B = 4πM + 1 B = μH
M = χH
©1999 E.A. Fitzgerald
B magnetic induction
χ magnetic susceptibility
H magnetic field strength (applied field)
M magnetization
2
H and B
• H has the possibility of switching directions when leaving the material;
B is always continuous
H
B
p
M
M
At p:
©1999 E.A. Fitzgerald
q
B H
M=0
H
At q: B
M
3
Maxwell and Magnetic Materials
• Ampere’s law ∫ H ⋅ dl = I = 0
• For a permanent magnet, there is no real current
flow; if we use B, there is a need for a fictitious
current (magnetization current)
• Magnetic material inserted inside inductor
increases inductance
⎛ 4π ⎞
Φ B = BA ~ 4πMA = 4πχHA = 4πχ ⎜
In ⎟ A
c
⎝
⎠
NΦ B (4π ) 2
=
n lAχ
L=
I
c
2
L increased by ~χ due to
magnetic material
Material Type
χ
Paramagnetic
+10-5-10-4
Diamagnetic
-10-8-10-5
Ferromagnetic +105
4
©1999 E.A. Fitzgerald
Microscopic Source of Magnetization
•
•
No monopoles
magnetic dipole comes from moving or spinning electrons
Orbital Angular Momentum
μ
μ is the magnetic dipole moment
r r
Energy = E = − μ ⋅ H = − μ H cos θ
I
A
e-
L
What is μ? For θ=0,
E = − μH ≈ −Φ B I since energy ~ LI 2 and for 1loop L =
ΦB
I
Φ B = ∫ ∫ H ⋅ dS ~ HA
∴ μH = Φ B I = HAI and ∴ μ = IA
e ω
A = πr 2
c 2π
e
μ = − ωr 2
2c
I =−
©1999 E.A. Fitzgerald
5
Microscopic Source of Magnetization
•
Classical mechanics gives orbital angular momentum as:
r r r
L = r × p = mr 2ω
e
eh
LQM = −
LZ = − μ B LZ
μL = −
Example for l=1:
2mc
2mc
eh ⎞
⎛
=
μ
⎜ B
⎟
E(H=0)
2mc
⎝
⎠
LZ = ml = −l,...,0,...l
0
-μBH
μs
Spin Moment
e
eh
S = − g0
S z = −g 0 μ B S Z
mc
2mc
1
S Z = mS = ±
g 0 = 2 for electron spin
2
μs = −
+μBH
+(1/2)μBg0H
Q.M .
E(H=0)
-(1/2)μBg0H
6
©1999 E.A. Fitzgerald
Total Energy Change for Bound Electron in Magnetic Field
•
Simple addition of energies if spin-orbital coupling did not exist
r r
E = − μ ⋅ H = μ B (LZ + g 0 S Z )H = μT H
But spin-orbit coupling changes things such that:
μT ≠ μ B (LZ + g 0 S Z ) = μ B J Z
QM definitions:
L = hLZ = hml
S = hS Z = hms
J = L+S
J = hJ Z
μ T = gμ B J Z
3 1 ⎡ S (S +1) − L(L +1) ⎤
g= + ⎢
⎥
2 2⎣
J ( J +1)
⎦
r
r
r
r
E = − μ T ⋅ H = gμ B J ⋅ H = gμ B J Z H
7
©1999 E.A. Fitzgerald
Total Energy Change for Bound Electron in Magnetic Field
•
Kinetic energy from Lorentz force has not been included
pH = −
e r r
r ×H
2c
Lorentz for circular orbit
(
)(
p2
e2 r r r r
Energy change =
=
r ×H ⋅ r ×H
2
2m 8mc
)
For the plane perpendicular to H and assuming circular orbit:
e2 2 2
e2
(
Energy change =
r H =
x 2 + y 2 )H 2
2
2
8mc
8mc
e2
2
2
2
(
)
∴ΔETOT = gμ B HJ Z +
+
H
x
y
8mc 2
Numbers:
μBH for
H=10-4 Gauss
=10-4 eV
for
H=10-4 Gauss
=10-9 eV
The Lorentz effect is minimal with respect to magnetic moment interaction, if it exists
8
©1999 E.A. Fitzgerald
Atoms with Filled Shells
• J=0 (L=0, S=0)
• Only Lorentz contribution
• Leads to diamagnetism
(
e2
2
2
2
ΔE =
H
x
+
y
8mc 2
)
Need to sum over all e- in atom:
ΔEatom
e2
2 2
2
(for a spherical shells)
H
r
=
∑
i
2
12mc
3 i
N ∂2E
χ=
V ∂H 2
1 ∂E
⎛
=
−
M
⎜
V ∂H
⎝
e2 N
χ =−
6mc 2 V
χ=
∂M ⎞
⎟
∂H ⎠
e2 N 2
∑i ri = − 6mc 2 V r Z i
2
e2 N 2
χ =−
r Zi
2
6mc V
~ -10-5
9
©1999 E.A. Fitzgerald
Atoms with Partially Filled Shells
• J not zero
• Need Hund’s rules from QM
– Fill levels with same ms to maximize spin
– maximize L (first e- goes in largest l)
– J=|L-S| for n<=(2l+1), J=|L+S| for n>(2l+1)
• Conventional notation: (2S +1) X J
• J=0 when L and S are not zero is a special case
– 2nd order effect--> perturbation theory
• Partially filled shells give atoms paramagnetic behavior
L
0
1
2
3
4
5
6
X
S
P
D
F
G
H
I
ΔE = gμ B HJ Z
(+10-2-10-3>10-9 eV for diamagnetic component)
10
©1999 E.A. Fitzgerald
d-shell (l = 2)
n
lx = 2, 1,
0,
-1,
-2,
1
2
3
4
5
6
7
8
9
10
J
S
L = |Σlx|
1/2
1
3/2
2
5/2
2
3/2
1
1/2
0
2
3
3
2
0
2
3
3
2
0
S
L = |Σlz|
J
1/2
1
3/2
2
5/2
3
7/2
3
5/2
2
3/2
1
1/2
0
3
5
6
6
5
3
0
3
5
6
6
5
3
0
5/2
4
9/2 J=|L-S|
4
5/2
0
7/2
6
15/2
8
15/2 J=L+S
6
7/2
0
3/2
2
3/2
0
5/2
4
9/2
4
5/2
0
Symbol
2
D3/2
F2
4
F3/2
5
D4
6
S5/2
5
D4
4
F9/2
3
F4
2
D5/2
1
S4
3
J=|L-S|
J=L+S
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
f-shell (l = 2)
n lz = 3, 2,
1
2
3
4
5
6
7
8
9
10
11
12
13
14
©1999 E.A. Fitzgerald
1,
0,
-1, -2,
-2,
2
F5/2
H4
4
I9/2
5
I4
6
H5/1
7
F6
8
S7/1
7
F6
6
H15/2
5
I8
4
I15/1
3
H6
2
F7/2
1
S0
3
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Figure by MIT OpenCourseWare.
11
Temperature Dependence of Paramagnetism
• Temperature dependence determined by thermal energy vs. magnetic
alignment energy (same derivation as for molecular polarizability in the case
of electric dipoles)
f =e
−U
k bT
=e
μZ
pE cosθ
k bT
μ fdΩ
∫
=
∫ fdΩ
Z
for electric dipoles;
f =e
−U
k bT
μH cosθ
=e
k bT
for magnetic dipoles;
For low H fields and/or low T,
μZ =
μ 2H
3kbT
=
μ B2 J 2 H
3kbT
N μ B2 J 2 H
M=
V 3kbT
N μ B2 J 2
χ=
V 3kbT
χ QM
1 N μ B2 g 2 J ( J + 1)
=
3V
kbT
χ∝
1
T
Curie’s Law
12
©1999 E.A. Fitzgerald
Effect of De-localized electrons on Magnetic Properties
•
Pauli Paramagnetism
– dues to the reaction of free e- to magnetic field
ΔE = g 0 μ B H
E(H=0)
M = − μ B (n+ − n− )
1
g + (E ) = g (E )
2
E(H=0)
1
g − (E ) = g (E )
2
μ = −g 0 μ B S
(n+ is the density of free electrons parallel to the H field)
g + (E ) =
1
g (E + μ B H )
2
ΔE = g 0 μ B H
1
g − (E ) = g (E − μ B H )
2
For exact solution, need
to expand about Ef for n+
and n-
Only e- near Fermi surface matter:
ΔE = g 0 μ B H
(n+ − n− ) = Δn ≈ g (EF ) ΔE
Δn ≈ g (EF )μ B H
©1999 E.A. Fitzgerald
2
M = μ B2 Hg (EF ), χ = μ B2 g (EF )
Note: Pauli paramagnetism has no T
dependence, whereas Curie paramagnetism has
1/T dependence
13
Effect of De-localized electrons on Magnetic Properties
• Landau paramagnetism
–
–
–
–
Effect of bands/Fermi surface on Pauli paramagnetism
F=qvxB for orbits
orbit not completed under normal circumstances
however, average effect is not zero
χ Land
1
= − χ Pauli
3
14
©1999 E.A. Fitzgerald
Ferromagnetism
• Most important but not common among elements
• Net magnetization exists without an applied magnetic field
• To get χ~104-105 as we see in ferromagnetism, most moments in material must be aligned!
• There must be a missing driving force
NOT dipole-dipole interaction: too small Edipole =
r
r
1 r r
−4
[
(
)(
)
]
μ
⋅
μ
−
3
μ
⋅
r̂
μ
⋅
r̂
~
10
eV
1
2
1
2
3
r
Spin Hamiltonian and Exchange
H spin = −∑ J ij S i S j
J ij ≡ exchange constant
Assuming spin is dominating magnetization,
() ( )(
)
()
r r r r
r r
r
1
H =− ∑
S R ⋅ S R ' J R − R ' − gμ B H ∑
SR
r r
r
2 R,R'
R
15
©1999 E.A. Fitzgerald
Exchange
E~-JS1S2
J negative, E~+S1S2--> Energy
if
J positive, E~-S1S2--> Energy
if
Fe, Ni, Co ---> J positive!
Other elements J is negative
Rule of Thumb:
r
interatomic distance
≡
> 1.5
2ra
2(atomic radius)
J is a function of distance!
16
©1999 E.A. Fitzgerald
Ferromagnetism
M
M (T ) ∝ (TC − T )
β ≈ 0.33 - 0.37
χ (T ) ∝ (TC − T )−γ
γ ≈ 1.3 -1.4
β
B=H+4πM
Br, Ms
Domain rotation
Irreversible boundary displacement
reversible boundary displacement
‘normal’ paramagnet
TC
Hc
H
T
Easy induction, “softer”
Magentic anisotropy
hardness of loop dependent on crystal direction
comes from spin interacting with bonding
17
©1999 E.A. Fitzgerald
Domains in Ferromagnetic Materials
B
N
S N S N
S
N S N S
M
Magnetic domain
Magnetic energy
1
= ∫ B 2 dV
8
Domain wall or boundary
Flux closure
No external field
18
©1999 E.A. Fitzgerald