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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



Mechanical oscillator model of electric dipole


Force acting on electric dipole
Potential energy of electric dipole in electric field
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
Ex t , z   E 0 coskz  t  0 
 E0 cosk /  z  t 0 
1 / c  k /    0 0
c  1 /  0  0  2.99792458 108 m/s (speed of light
in vacuum)
In dielectric material: ε(≠ε0), μ0
E x t , z   E0 cosk /  z  t 0 
1 / c'  k /   0
c'  1 / 0 
1 /  0 0
 / 0

c
Speed of light in material 
n n : refractive index
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Electromagnetic wave in
dielectric material

Electromagnetic wave in dielectric
material propagates with slower speed
c’ than that in vacuum c.
c'  c / n, n : refractive index
n   /  0 ,    0 1     n  1, c'  c

Measurement of n provides ε (orχ),
which describes the electric property of
a material. (Optical measurement)
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Phasor representation

Waves can be represented by complex
exponential function instead of real
trigonometric function.
EXERCISE:

Real trigonometric function

Complex exponential function (Phasor rep.)
Ex t , z   E 0 coskz  t  0 
~
~
Ex t , z   E 0 exp ikz  t 
1 ~
~
~*
Ex t , z   Re Ex t , z   Ex t , z   Ex t , z 
2
~
E 0  E0 exp i0  : complex amplitude

 

NOTE: “~” denotes phasor representation, and therefore it is complex.
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2. Electric dipole interaction:
Force and potential energy

Force acting on charge and potential
energy of electric dipole moment
Force acting on charge
F  QE
Electric dipole moment and Polarization
p  Qd Electric dipole moment
p Q Electric
P
 ex
polarization
Sd S
Potential energy
U   Q Ex0   Q E x0  d 
  E Qd    E  p
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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.
p  Qr
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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
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Relation between electric dipole moment
of atom and electric polarization
p  Npa  naVpa : total electric dipole moment
p
P   na pa : relation between P and pa
V
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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
pa  QR
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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”
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Oscillatory motion of electron cloud

Motion of the center of the electron
cloud ⇒ Damped harmonic oscillation
Equation of motion
d2
d
M 2 R  2M R  KR  QE0 cos t 
dt
dt
Set z=0, φ0=0 for simplification
Phasor representation
d2 ~
d ~
~
~
M 2 R  2M R  KR  QE0 exp  it 
dt
dt
~
R  Re R

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Solution of damped oscillation

Equation of motion (phasor rep.)
d2 ~
d ~
~
~
M 2 R  2M R  KR  QE0 exp  it 
dt
dt
EXERCISE: Solve the above differential equation.

Solution (phasor rep.)
~
~
R t   R0 exp  it  
~
R0 
Q/M
 2  02  2i
Q/M
~
E
0 exp  it 
2
2
  0  2i
~
E0
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Electric dipole moment of an atom,
polarization, susceptibility and permittivity

Electric dipole moment of an atom (Phasor)
~
~
pa t   QR0 exp  it  

Electric polarization (Phasor)
~
~
P   0 ~E t   na ~
pa
 na Q 2 / M ~
 na Q 2 / M
 2
E0 exp  it   2
2
2
  0  2i

 Q2 / M
~
E0 exp  it 
2
2
  0  2i
  0  2i
~
E t 
Electric susceptibility and permittivity (Phasor)
 na Q 2 /  0 M 
~

 2  02  2i
2

n
Q
/  0 M  
~
~
a
   0 1      0 1  2

2




2
i


0


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Refraction index and electric susceptibility

Relation between refraction index and
electric susceptibility
n~  ~ /  0
n~ 2  ~ /  0  1  ~  n~ must be complex.
~  n'in" , ~   'i "
Let n
 n'2 n"2  1   ' : real part
the n 
2n' n"   " : imaginary part
EXERCISE: Derive the above relation between n’, n” and χ’, χ”.
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Real and imaginary parts of n and χ

Electric susceptibility (Γ≪ω0)
 na Q 2 /  0 M 
~
   'i "  2
  02  2i
02 S 02   2
S0 0    / 2
Real part :
' 2

2
2
2
  02  4 2 2   0   
202 S
S0  / 2
na Q 2
Imaginary part :  " 

, where S 
2
2
2
2 2
2 2
M 002
  0   
  0  4 







Refractive index (n’≃1, n”≪1)
S0 0    / 4
0  
1

1


4 0 MV0   0 2   2
  0 2   2
S0 / 4
1

Imaginary part : n" 


  0 2   2 4 0 MV0   0 2   2
Real part :
n'  1 
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Refractive index (non-dimensional)
Graph of refractive index
n’ : real part
n” : imaginary part
Angular frequency (rad/s)
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4. Absorption and dispersion of
light in material

What are the real and imaginary parts
of refractive index?


Electric field (phasor rep.)
  n~
~
~
~

Ex t , z   E0 exp ik /  z  t  E0 exp i  z  t 
Replace n by n’+in”
  n'in"
~
~

E x t , z   E0 exp i 
z  t 
 c

  c
  n'
~

 n"
 E0 exp i  z  t   exp  
c


 c


z

Propagating wave
Spatial damping
n’ ⇒traditional
n” ⇒absorption
refractive index
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Absorption of light

Absorption: n” describes damping of
wave by dielectric.
Vacuum
Dielectric
D
Vacuum
Damping of electric field
during D
 n" 
exp  
D
 c

Damping of light
intensity I (∝E2)
 2n" 
exp  
D   exp  AD 
c


 D 

 exp  
 d 
 opt 
2n"
A
: Absorption coefficien t
c
1
c
d opt  
: Optical depth
A 2n"
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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.
Snell' s law :
n1 sin 1  n2 sin  2
n1, 2 : refracttiv e index of dielectric 1, 2
1,2 : angles of incidence and refraction
in dielectric 1, 2
When n' blue   n' red ,  b   r .
EXERCISE:
Prove the above inequality.
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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
n'2 n"2  1   '
2n' n"   "
Atomic property
Electric dipole moment
of atom
ω0: Resonance
frequency
Γ : Damping constant
'
S0 0    / 2
  0 2   2
"
S 0  / 2
  0 2   2
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Summary


Review of electromagnetic wave
Electric dipole interaction



Mechanical oscillator model of electric dipole


Force acting on electric dipole
Potential energy of electric dipole in electric field
Lorentz model and refraction index
Absorption and dispersion of light in material

Absorption and refraction
22