Lecture 35

Lecture 35 - EEE
352—Lecture
35
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Unformatted text preview: EEE
352—Lecture
35
 Magne1c
Materials
 Magne&c
force
microscope
image
of
the
bits
in
 IBM’s
20
Gb/in2
memory.
 Philip
Anderson
 Sir
Nevill
Mo>
 John
van
Vleck
 Nobel
prize
in
Physics,
1977,
for
“for
their
fundamental
theore1cal
 inves1ga1ons
of
the
electronic
structure
of
magne1c
and
disordered
 systems”
 Magne1c
Forces
and
Dipoles
 We
have
already
dealt
with
magne1c
forces
in
trea1ng
the
Hall
effect
on
 electrons
and
holes.

Let
us
review
some
of
that
work
and
 electromagne1cs.
 Magne&c
flux
density
 Tesla
=
Webers/m2
 Flux
in
Webers
 Area
in
m2
 Force
created
by
velocity
(or
current)
 Field
created
by
a
current:
 ∫ H • dl = ∫∫ ∇ × H • ndA = ∫∫ J • ndA = I Hφ = I Ampere / meter 2πr € Magne1c
Suscep1bility
 Consider
a
cylinder
of
dielectric
 material
with
a
cylindrical
hole
(of
 cross‐sec1onal
area
of
dA)
cut
 down
the
axis
of
the
cylinder.
 The
hole
is
filled
with
air,
just
as
the
 medium
around
the
cylinder.
 Magne1c
Suscep1bility
 In
the
cavity
 while
in
the
cylinder
 Since
the
two
values
of
the
flux
density
are
equal
(by
construc1on),
 then
the
field
intensi1es
are
related
by
 Hence,
the
necessary
current
(circula1ng
around
the
hole)
is
 I ′ = ( H cavity − H ) dl = (µr − 1) Hdl and
the
magne1za1on
(dipole)
is
 € Here,
M
is
the
magne1za1on.

(This
corresponds
to
the
“polariza1on”
 in
dielectric
material—polariza1on
arises
from
charge
dipoles,
while
 magne1za1on
arises
from
current—magne1c
dipoles)
 nucleus
 d
 Charge
separa1on
produces
dipoles,
 of
value
 electrons
 p = qd P = Np m
 Current
loops
produce
magne1c
 dipoles,
of
value
 I
 € µm = IA M = Nµm € B = µ0 ( H + M ) M = χmH µ = µ0 (1 + χ m ) = µ0µr D = ε0 ( E + P ) P = χe E ε = ε0 (1 + χ e ) = ε0εr € € There
is
a
1:1
duality
between
electric
and
magne1c
fields
(E
and
H),
 just
as
between
voltage
and
current.

That
is,
E
and
H
are
duals,
B
and
D
 are
duals,
flux
and
charge
are
duals.
 In
the
world
of
special
rela1vity,
not
everything
is
rela&ve!

Charge
and
 flux
are
not
rela1ve,
and
do
not
depend
upon
the
coordinate
system,
or
 its
mo1on.
 In
dielectric
materials,
we
had
several
types:
normal
dielectric
 materials,
piezoelectrics,
ferroelectrics,
and
pyroelectrics.
 Similarly,
there
are
several
types
of
magne1c
materials:
 
diamagne1c
 
paramagne1c
 
ferromagne1c
 
an1‐ferromagne1c
 € χm < 1 χ m > 1, small χ m >> 1 
ferrimagne1c
 There
are
also
piezomagne1c
and
pyromagne1c,
but
there
use
is
 very
small,
and
we
will
not
discuss
them
further.
 Diamagne1sm
 A
material
is
diamagne1c
when

 EXAMPLE:
orbital
electron
diamagne1sm
 eω 0 2 µm = IA = − (πr ) 2π orbi&ng
frequency
 orbit
radius
 Suppose
we
apply
a
small
magne1c
field
 € V = − ∫ Edl = E =− r dB 2 dt dΦ d , 2πrE φ = (πr 2 Bz ) dt dt € Diamagne1sm
 The
change
in
angular
momentum
is
 Then
 Larmor
frequency
 (equal
½
cyclotron
frequency)
 Now,
the
magne1za1on
is
given
as
 e 2r 2 M = Nµm = − B 4m = χmH e 2r 2 B = µ0 H − µ0 B 4m µ0 µ= < µ0 22 er 1+ µ0 4m € Diamagne1sm
arises
from
a
repulsive
interac1on—the
applied
field
 slows
down
the
orbits
causing
a
field
which
opposes
the
applied
 field.
 This
can
be
used
to
levitate
diamagne1c
media,
because
of
the
 opposing
forces.
 h>p://www.otherpower.com/diacglev.html
 h>p://www.hfml.science.ru.nl/froglev.html
 Paramagne1sm
 A
material
is
paramagne1c
when

 If
the
orbits
are
free
to
rotate,
then
they
will
rotate
to
aid
the
 magne1c
field.

We
had
previously
 In
an
atom
 so
that
 Bohr
magneton
β Paramagne1sm
 Another
source
of
magne1c
moment
is
the
spin
of
the
electrons:
 The
factor
gs
arises
because
there
is
a
strong
COUPLING
between
 ORBITAL
and
SPIN
angular
momentum.
 EXAMPLE:
we
calculate
the
spin
contribu1on
to
the
total
magne1za1on.
 The
energy
change
when
the
field
is
applied
is
 Each
electron
state
is
weighted
by
a
factor
determined
by
its
gain/ loss
in
energy.
 Electron
Spin
 Discovered
experimentally
by
S.
A.
Goudsmit
 and
George
Uhlenbek
in
1925.
 They
found
that
the
electron
had
an
 addi1onal
angular
momentum,
which
 possessed
only
two
values,

 George
Uhlenbeck
 S.
A.
Goudsmit
 Pauli
realized
the
importance
of
this
extra
angular
 momentum
and
postulated
the
exclusion
principle,
in
 which
a
real
space
quan1zed
state
could
have
only
two
 electrons,
with
opposite
spin,
present.

This
led
to
the
 quantum
sta1s1cs
of
the
Fermi‐Dirac
distribu1on.
 Only
Pauli,
of
this
group,
received
the
Nobel
Prize!

 This
was
awarded
for
the
exclusion
principle.
 Wolfgang
Pauli
 Nobel
prize
in
physics,
1945
 Electron
Spin
 The
electron
spin
is
assumed
to
arise
from
the
electron
rota1ng
around
its
own
 axis.

But,
which
axis?

It
turns
out
that
it
is
whichever
axis
is
being
measured!

(But,
 one
can
only
measure
one
axis
at
a
1me.)
 The
two
spin
states
can
be
separated
by
a
magne1c
field,
and
the
transi1on
 between
the
two
states
is
used
in
electron
spin
resonance
studies
of
materials.
The
 spliing
is
called
the
Zeeman
effect.
 In
a
magne1c
field
(in
z‐direc1on,
eqn.
5.97)
 Pieter
Zeeman
 Nobel
Prize
in
Physics,
1902
 (shared
with
Hendrik
Lorentz
 for
studies
in
magne&sm)
 g
is
a
“fudge”
factor,
=2
for
free
electrons
 This
quan1ty
is
known
as
the
Bohr
magneton.

While
it
is
associated
with
 the
Zeeman
effect,
it
was
discovered
by
Bohr
and
carries
his
name,
not
 Zeeman’s.
 Now,
let
us
return
to
the
idea
of
the
popula1on
of
each
spin.

The
energy
 shim
is
due
to
the
Zeeman
effect,
as
we
saw
earlier.

What
we
seek
is
the
 spin
polariza1on,
or
more
properly,
its
average
over
all
of
the
angles
in
the
 solid.
 Now,
in
a
metal,
most
of
the
spins
cannot
turn
in
the
magne1c
field,
since
 those
states
are
occupied.

Instead,
we
rely
upon
the
change
in
energy
 1 ± gβB ≈ ± βB 2 € Parallel
to
the
field
 Opposite
to
the
field
 2βB 1F 1 N↑ = ∫ ρ( E + βB)dE = 2 − βB 2 1F 1 N↓ = ∫ ρ( E − βB)dE = 2 βB 2 E E F + βB € E 1 ∫ ρ( E )dE ~ 2 0 EF ∫ ρ( E )dE + 0 EF βB ρ( E F ) 2 βB ρ( E F ) 2 E F − βB 1 ∫ ρ( E )dE ~ 2 0 ∫ ρ( E )dE − 0 € Now,
the
magne1za1on
is
just
 M = β ( N↑ − N↓ ) ~ β 2 Bρ( E F ) From
our
previous
discussions
of
the
density
of
states
and
the
Fermi
 energy,
we
have
 € ρ( E ) = KE 1 / 2 F ρ ( E ) dE 2 3 N= ∫ = K ∫ ρ ( E ) dE = KE F / 2 ( E − E F ) / kB T 3 0 1+ e 0 3N 3N ρ( E F ) = ≡ 2 E F 2 kB TF ∞ E and
 € 3Nβ 2 M= B 2 k B TF € Now,
the
magne1za1on
is
given
as
 3Nβ 2 M= B 2kB T = χmH 3Nβ 2 B = µ0 H + µ0 B 2kB T µ0 µ= > µ0 3Nβ 2 1− µ0 2kB T € New Japanese Maglev---581 km/hr The French TGV has demonstrated 574 km/hr—on track! ...
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This note was uploaded on 10/28/2009 for the course EEE 352 taught by Professor Ferry during the Fall '08 term at ASU.

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