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Introduction to materials physics #3
Week 3: Electric dipole interaction
1
Chap. 1-2: Table of contents
Review of electromagnetic wave Electric dipole interaction
Force acting on electric dipole Potential energy of electric dipole in
electric field Mechanical oscillator model of electric
dipole Lorentz model and refraction index
Absorption and dispersion of light in material Absorption and refraction
2
1. Review of electromagnetic wave: Electromagnetic waves in vacuum VS. dielectric material In vacuum: ε0, μ0
In dielectric material: ε(≠ε0), μ0
00
00
00
//1
/coscos,
kc
tzkEtkzEztEx
in vacuum) light of (speed m/s 102.99792458/1 8
00 c
/
/1/1'
/'/1
/cos,
0
000
0
00
n
cc
kc
tzkEztEx
index refractive:
materialin light of Speed
n 3
Electromagnetic wave in dielectric material Electromagnetic wave in dielectric
material propagates with slower speed c’ than that in vacuum c.
Measurement of n provides ε (orχ), which describes the electric property of a material. (Optical measurement)
ccnnnncc
' ,1 1 ,/ index refractive : ,/'
00
4
Phasor representation Waves can be represented by
complex exponential function instead of real trigonometric function. Real trigonometric function
Complex exponential function (Phasor rep.)
00cos, tkzEztEx
amplitudecomplex :exp~
,~
,~
2
1,
~Re,
iexp~
,~
000
*
0
iEE
ztEztEztEztE
tkzEztE
xxxx
x
NOTE: “~” denotes phasor representation, and therefore it is complex.
EXERCISE:
5
2. Electric dipole interaction:Force and potential energy Force acting on charge and
potential energy of electric dipole moment
pE
QdE
dxEQExQU 00
Potential energy
Electric dipole moment and Polarization
xS
Q
Sd
Q
ep
P
dp
Force acting on charge
EF Q
Electric dipole moment
Electric polarization
6
Electric dipole moment Electric dipole moment is a pair of
two positive and negative charges with the same magnitude separated with the displacement vector r.
rp Q
7
Electric polarization and electric dipole moment of atoms (or molecules) Electric polarization consists of electric
dipole moments of atoms.
To know electric dipole moment of a single atom is equivalent to know electric polarization
8
Relation between electric dipole moment of atom and electric polarization
aaa
aaa
and between relation :
moment dipole electric total:
pPpp
P
ppp
nV
VnN
9
3. Mechanical oscillator model of electric dipole:Electric dipole moment of an atom induced by external electric field
An atom consists of a positively charged nucleus and negatively charged electron cloud. If external electric field exists, the nucleus and the center of the electron cloud are displaced. ⇒ electric dipole moment
Without E field With E field
Rp Qa
10
Electric dipole moment as an mechanical oscillator: Lorentz model Electric dipole moment of an atom
can be regard as a mechanical oscillator.
○ Stronger electric field displaces the electron cloud farther. ⇒ “Spring”○ Inertia of the electron cloud ⇒ “Mass”
11
Oscillatory motion of electron cloud
Motion of the center of the electron cloud ⇒ Damped harmonic oscillation
Equation of motion
tQEKRRt
MRt
M cosd
d2
d
d02
2
Set z=0, φ0=0 for simplification
Phasor representation
RR
tEQRKRt
MRt
M~
Re
iexp~~~
d
d2
~
d
d02
2
12
Solution of damped oscillation Equation of motion (phasor rep.)
Solution (phasor rep.)
tEQRKRt
MRt
M iexp~~~
d
d2
~
d
d02
2
020
20
020
20
~
i2
/~
iexp~
i2
/iexp
~~
EMQ
R
tEMQ
tRtR
EXERCISE: Solve the above differential equation.
13
Electric dipole moment of an atom, polarization, susceptibility and permittivity
Electric dipole moment of an atom (Phasor)
Electric polarization (Phasor)
Electric susceptibility and permittivity (Phasor)
tEMQn
tEMQnpntEP
~
i2
/iexp
~
i2
/
~~~~
20
2
2a
020
2
2a
aa0
tEMQ
tRQtp
iexp~
i2
/iexp
~~02
02
2
0a
i2
/1~1~
i2
/~
20
20
2a
00
20
20
2a
MQn
MQn
14
Refraction index and electric susceptibility
Relation between refraction index and electric susceptibility
complex. bemust ~ ~1/~~/~~
02
0
nn
n
partimaginary : ""'2
part real : '1"'n the
"i'~ ,"i'~Let 22
nn
nn
nnn
EXERCISE: Derive the above relation between n’, n” and χ’, χ”.
15
Real and imaginary parts of n and χ
Electric susceptibility (Γ≪ω0)
Refractive index (n’≃1, n”≪1)
200
2a
220
0
22220
2
20
220
00
22220
2
220
20
20
20
2a
where,2/
4
2" :partImaginary
2/
4' :part Real
i2
/"i'~
M
QnS
SS
SS
MQn
22000
220
0
220
0
0022
0
00
4
14/" :partImaginary
4
11
4/1' :part Real
MV
Sn
MV
Sn
16
Graph of refractive index
Angular frequency (rad/s)
Refr
act
ive in
dex (
non-d
imensi
onal)
n’ : real part
n” : imaginary part
17
4. Absorption and dispersion of light in material What are the real and imaginary
parts of refractive index? Electric field (phasor rep.)
Replace n by n’+in”
~
iexp~
/iexp~
,~
00
tzc
nEtzkEztEx
zc
ntz
c
nE
tzc
nnEztEx
"exp
'iexp
~
"i'
iexp~
,~
0
0
Spatial damping n” ⇒absorption
Propagating wave n’ ⇒traditional refractive index
18
Absorption of light Absorption: n” describes damping
of wave by dielectric.Vacuum VacuumDielectric
D
D
c
n "exp
Damping of electric field during D
depth Optical : "2
1
tcoefficien Absorption : "2
exp
exp"2
exp
opt
opt
n
c
Ad
c
nA
d
D
ADDc
n
Damping of light intensity I (∝E2)
19
Dispersion: separation of colors
n’ is a function of ω. ⇒ Refraction is different among colors. In most cases, ω0≫ω. ⇒ n’ (ωblue)>n’
(ωred) Blue ray bends more deeply that red
ray does.
2 1, dielectricin refraction and incidence of angles :
2 1, dielectric ofindex erefracttiv : sinsin
:law sSnell'
1,2
2,1
2211
n
nn
. ,''When rbredblue nn
EXERCISE: Prove the above inequality.20
How does one probe property of atoms from optical measurement? Mutual relation among optical,
electric and atomic properties
Optical property Refractive index n’ : Refraction n” : Absorption
Electric property (Dielectricity) Electric susceptibility χ’ : Real part χ” : Imaginary part
Atomic property Electric dipole moment of atom ω0: Resonance frequency Γ : Damping constant
""'2
'1"' 22
nn
nn
220
0
220
00
2/"
2/'
S
S
21
Summary
Review of electromagnetic wave Electric dipole interaction
Force acting on electric dipole Potential energy of electric dipole in
electric field Mechanical oscillator model of electric
dipole Lorentz model and refraction index
Absorption and dispersion of light in material Absorption and refraction
22