Download Chapter 6 Magnetic Fields in Matter 6.1 Magnetization

Chapter 6 Magnetic Fields in Matter
6.1 Magnetization
6.1.1 Diamagnets, Paramagnets, Ferromagnets
All magnetic phenomena are due to electric charges in motion.
r
r
Electric polarization p is almost always in the same direction as E while the
r
magnetic polarization can be either parallel to or opposite to B .
E
+
-
p
B
m
B
m
or
B
m
magnetic polarization Æ magnetization
r
a magnetization parallel to B : paramagnets (paramagnetic, paramagnetism)
r
a magnetization opposite to B : diamagnets (diamagnetic, diamagnetism)
spontaneous magnetization (retain their magnetization even in a zero field):
ferromagnets, antiferromagnets, and ferrimagnets (ferromagnetic, ferromagnetism,
antiferromagnetic, antiferromagnetism)
Magnetic Response
Itinerary s- or p-electrons (traveling
in the metal)
Local electrons (around nucleus)
Diamagnet, Larmor
or Langevin
Paramagnet, Curie
(d-electrons)
Diamagnet, Landau
diamagnetism
Paramagnet, Pauli
paramagnetism
Below Tc
Ferro
Antiferro
Ferri
Reminding again:
What’s the difference between an electric polarization and a magnetization?
Electric Dipole: electric field Æ use force to separate the two opposite charges ± e
r r
r r
Æ waste electric displacement D , D = ε 0 E + P
Magnetic Dipole: magnetic field Æ use torque to align the pointing direction of a
magnetization (paramagnetism) Æ once you align it you gain magnetic fields
r
1 r r
H=
B−M
µ0
6.1.2 Torques and Forces on Magnetic Dipoles
r r
The torque on electric dipole under a uniform electric field: τ = p × E .
E
+
(Coulomb force)
What’s the torque on the magnetic dipole when placing in a magnetic field?
Go back to the current model and use the Lorentz force:
r r
r
The Lorentz force law: Fmag = ∫ I dl × B
(
F
)
In a uniform magnetic field:
B
θ
I
r
r
r
r
F
θ
r
τ = 2 * F sin θ = a(IBa )sin θ = Ia 2 B sin θ → τ = Ia × B = m × B
a
2
The torque to line the dipole up accounts for paramagnetism. (Thermal energy breaks
up the ordering alignment of dipoles parallel to the magnetic field.)
In a uniform field, the net force on a current loop is zero:
r r
r r
F = I ∫ dl × B = I ∫ dl × B = 0
( )
In a nonuniform magnetic field:
B
F
For example, a loop is situated above a solenoid with same directions of current flow.
(
)
r
r r
r
F = ∇ m ⋅ B for an infinitesimal loop with dipole moment m .
Model:
N
S
+
-
Ampere’s model
Gilbert model
r
You can use Gilber model (when ∇ × H = 0 ), treat magnetic dipoles as electric
dipoles, and obtain the magnetic fields. It gives a correct field of dipole but incorrect
physical concept.
6.1.3 Effect of a Magnetic Field on Atomic Orbits
Classical view of diamagnetism:
Model: An electron with charge of − e orbiting around a nucleus with charge of
+e.
e
ve
2πR
, and the current is I = =
.
v
T 2πR
z
r
r
ve
evR
The orbital dipole moment is m = I ⋅ a = −
πR 2 zˆ = −
zˆ .
2πR
2
-e, v
m ∝ −v
The period of the orbital motion is T =
Under zero magnetic field, the electrostatic force is used for circular motion.
F =−
1 e2
v2
e2
v2
=
−
Æ
=
m
− − − (1)
m
e
e
4πε 0 R 2
R
4πε 0 R 2
R
1
When we turn on the field, a Lorentz force is
introduced to change speed of orbital motion.
z, B
Lorentz force
-e, v
e2
v2
+
e
v
B
=
m
− − − (2)
e
4πε 0 R 2
R
1
me
(v + v )(v − v ) ≅ me 2v ∆v (as v is close to v )
R
R
2 2 r
r
eRB
eR
e R
∆v =
Æ ∆m = − ∆vzˆ = −
B
2me
2
4me
(2) – (1) Æ ev B =
The negative magnetization means diamagnetic polarization.
Diamagnetism exists in all atoms. – How can we observe it in paramagnetic
materials?
B
No Quantum Mechanis Æ No Para- or Diamagneetism?
(See Feynman’s lecture)
Hamiltonian for a charged particle moving in a field: Diamagnetism
r
r
r
r
Kinetic momentum: pkin = hk , Field momentum: p field = qA , while the conjugate
r
r
r
r
momentum of x is pcanonical = pkin + p field . From energy point of view, we have
(
)
r2
r
r
r
r
2
(
pcan − p field )
p 2 kin
pcan − qA
H=
+V =
+V =
+V .
2m
2m
2m
r r
r
r
1
1
1r r
Pˆ 2
H=
+V =
( pi + eA(ri ))2 + V =
( pi − e ri × B) 2 + V
∑
∑
2m
2m i
2m i
2
r
r r
assume B = Bzˆ Æ r × B = iˆyB − ˆjxB
r2
1
e
e2 B 2
2
2
(
)
Hˆ =
p
+
V
−
p
y
−
p
x
B
+
( xi + yi )
∑
∑
∑
i
ix
iy
2m i
2m i
8m i
r2
1
e
e2 B 2
2
2
(
)
Hˆ =
p
+
V
+
xp
−
yp
B
+
( xi + yi )
∑
∑
∑
i
iy
ix
2m i
2m i
8m i
r2
1
e
e2 B 2
2
2
Hˆ =
p
+
V
+
l
B
+
( xi + yi )
∑
∑
∑
i
iz
2m i
2m i
8m i
r r e2 B 2
r2
1
e
2
2
Hˆ =
p
+
V
+
l
( xi + yi )
∑
∑
∑
i
i ⋅B+
2m i
2m i
8m i
B
After including spin term:
(
q
)
-e
r r e2 B 2
r2
1
e r
2
2
Hˆ =
p
+
V
+
L
+
g
S
⋅B+
( xi + yi )
∑
∑
i
0
2m i
2m
8m i
The diamagnetic term:
e2 B 2
8m
∑ (x
2
i
le
+ yi )
2
i
Nµ d
e2
∂
e B
2
2
2
E=−
(
x
+
y
)
χ
=
=
−
µ
N
ri
,
∑
∑
i
i
d
0
∂B
4m i
H
6m i
r
r
∂
e r
µp = − E = −
L + g0 S
2m
∂B
2
µd = −
(
He
−6
χ(10 cm /mol) -1.9
3
)
Ne
Ar
Kr
Xe
-7.2
-19.4
-28
-43
Fχ(10−6cm3/mol) -9.4
Cl-
Br-
Li+
Na+
-24.2
-34.5
-0.7
-6.1
6.1.4 Magnetization
Matter becomes magnetized under an applied field.
M
m
r
We describe the state of magnetic polarization by volume magnetization, M .
r
M = magnetic dipole moment per unit volume
r
m = total magnetic moment
Diamagnetism and paramagnetism are extremely weak in comparison with
atom
ferromagnetism.
Thermal energy
No exchange interaction
r
r
M Ferromagnetism ~ 105 ∗ M Parramagnetism
Exchange interaction
Estimate the factor by examining the temperature
dependence of magnetic susceptibility
Where does the atomic moment come from? – spin and orbital moment
Reminding:
r
r µ m
r
× rˆ µ0 m sin θ ˆ
0
ˆ
=
m = mz , A =
φ
4π r 2
4π r 2
r
r
Bdipole = ∇ × A =
rˆ
1
∂
2
r sin θ ∂r
rθˆ
∂
∂θ
0
0
Exercise: 6.4
r
θ
m
ˆ
r sin θφ
∂
µm
= 0 3 2 cosθrˆ + sin θθˆ
4πr
∂φ
µ0 m sin θ
r sin θ
4π r 2
(
)