Electric Dipole Interaction in Materials Physics

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Introdu
ction 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
 
A
bsorption and dispersion of light in material
Absorption and refraction
 
2
 
1. Review of electromagnetic wave:
Electromagnetic waves in vacuum VS.
dielectric material
 
In vacuum:
 
ε
0
, 
μ
0
 
 
 
 
In dielectric material: 
ε
(
ε
0
), 
μ
0
 
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
 (or
χ
),
which describes the electric property of
a material. (Optical measurement)
 
4
 
Phasor representation
 
Waves can be represented by complex
exponential function instead of real
trigonometric function.
Real trigonometric function
 
Complex exponential function (Phasor rep.)
NOTE: “~” denotes phasor representation, and therefore it is complex.
 
EXERCISE:
 
5
 
2. Electric dipole interaction:
F
orce and potential energy
 
Force acting on charge and potential
ener
gy of electric dipole moment
 
Potential energy
 
Electric dipole moment and Polarization
 
Force acting 
on
 charge
 
Electric dipole moment
 
Electric
   
 polarization
 
6
 
Electric dipole moment
 
Ele
ctric dipole moment is a pair of two
positive and negative charges with the
same magnitude separated with the
displacement vector 
r
.
 
7
 
E
lectric polarization and electri
c 
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 ele
ctric dipole moment
of atom and electric polarization
 
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
 
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 t
he center of the electron
cloud
 
 
Damped harmonic oscillation
Equation of motion
 
Set 
z
=0, 
φ
0
=0 for simplification
Phasor representation
 
12
 
Solution of damped oscillation
 
Equation of motion (phasor rep.)
 
 
Solution (phasor rep.)
 
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)
 
14
 
R
efraction index and electric susceptibility
 
Relation between refraction index and
electric susceptibility
 
EXERCISE: Derive the above relation between 
n’, n” 
and 
χ’, χ”
.
 
15
 
Real and imaginary parts of n and 
χ
 
Electric susceptibility (
Γ
ω
0
)
 
 
 
 
 
 
Refractive index (
n’
1, 
n”
1)
 
16
 
Graph of refractive index
 
Angular frequency (rad/s)
 
Refractive index (non-dimensional)
 
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”
 
Spatial damping
    
n” 
absorption
 
Propagating wave
    n’
 
traditional
             refractive index
 
18
 
Absorption
 
of light
 
Absorption: 
n”
 describes damping of
wave by dielectric.
 
Vacuum
 
Vacuum
 
Dielectric
 
D
 
Damping of electric field
     during 
D
 
Damping of light
  
intensity 
I
 (
E
2
)
 
19
 
Dispersion: separation of colors
 
n’ 
 is a function of 
ω
. 
 
Refraction is
different among colors.
In most cases, 
ω
0
ω
. 
 
n’ 
(
ω
blue
)>
n’ 
(
ω
red
)
Blue ray 
bends more deeply that red ray does.
 
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
      
ω
0
: Resonance
                     frequency
      Γ 
: Damping constant
 
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
 
A
bsorption and dispersion of light in material
Absorption and refraction
 
22
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Dive into the world of electric dipole interaction in materials physics, exploring topics such as electromagnetic wave properties, force acting on electric dipoles, potential energy calculations, and the representation of waves through phasors. Learn about the differences between electromagnetic wave propagation in vacuum versus dielectric materials, refractive indices, and the measurement of electric properties in materials. Discover the intricacies of electric dipole moments, polarization, and potential energy calculations within an electric field.


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  1. Introduction to materials physics #3 Week 3: Electric dipole interaction 1

  2. 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

  3. 1. Review of electromagnetic wave: Electromagnetic waves in vacuum VS. dielectric material In vacuum: 0, 0 ( ) ( ) ( ) 0 / / 1 = k c 2.99792458 / 1 0 0 = = c = k + z , cos Ex t z E kz t 0 0 = + cos / E t 0 0 0 8 10 (speed m/s of light in vacuum) ) / cos , 0 t z k E z t Ex = In dielectric material: ( 0), 0 ( ) ( / ' / 1 k c = + 0 0 / 1 c ( n ) / 1 = = = 0 0 ' c Speed of material in light 0 n / : refractive index 3 0

  4. Electromagnetic wave in dielectric material Electromagnetic wave in dielectric material propagates with slower speed c than that in vacuum c. n n c c = = , / 0 0 = ' / , refractive : index ( ) + 1 n ' , 1 n c c Measurement of n provides (or ), which describes the electric property of a material. (Optical measurement) 4

  5. Phasor representation Waves can be represented by complex exponential function instead of real trigonometric function. Real trigonometric function ( ) ( , + = t kz E z t Ex ) 0cos 0 Complex exponential function (Phasor rep.) ( ) ( ) ( ) ( ) ( ) complex : exp 0 0 0 i E E = ~ E ~ E = , exp i t z kz t 0 x EXERCISE: 1 ~ E ~ E ~ E ( ) ( ) = = + * x , Re , , , E t z t z t z t z x x x 2 ~ amplitude NOTE: ~ denotes phasor representation, and therefore it is complex. 5

  6. 2. Electric dipole interaction: Force and potential energy Force acting on charge and potential energy of electric dipole moment Force acting on charge F Q = E Electric dipole moment and Polarization Q p P = = p d Electric dipole moment Q Electric polarization e x Sd S Potential energy ( E ) ( E ) ( p ) = + + U Q Ex Q E x d 0 0 ( ) = = Qd 6

  7. 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 r Q 7

  8. 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

  9. Relation between electric dipole moment of atom and electric polarization = = p p p total : electric dipole moment N n V a a a p V = = P p P p relation : between and n a a a 9

  10. 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 = p R Q a 10

  11. 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

  12. Oscillatory motion of electron cloud Motion of the center of the electron cloud Damped harmonic oscillation Equation of motion d 2 2 d ( ) t + + = QE 2 cos M R M R KR 0 d d t t Set z=0, 0=0 for simplification Phasor representation ~ d = 2 d d ~ ~ ~ ( ) + + = 2 exp i M R M R K R Q E t 0 2 d t Re t ( ) R ~ R 12

  13. Solution of damped oscillation Equation of motion (phasor rep.) 2 d d ~ ~ ~ ~ ( ) t + + = i 2 exp M R M R K R Q E 0 2 d d t t EXERCISE: Solve the above differential equation. Solution (phasor rep.) / Q M + ~ R ~ R ~ E ( ) t ( ) ( ) = = exp i exp i t t 0 0 2 2 0 i 2 / Q M + ~ R ~ E = 0 0 2 2 0 i 2 13

  14. Electric dipole moment of an atom, polarization, susceptibility and permittivity Electric dipole moment of an atom (Phasor) ( ) ( ) t R Q t p 2 / + Q M i 2 ~ ~ ( ) t ~ = i = i exp exp E a 0 0 2 2 0 Electric polarization (Phasor) ( ) Q n i 2 0 + ~ ~ ~ ~ = = P E t n / p 0 a a 2 2 / M n Q 2 0 M i 2 + ~ ~ ( ) ( ) t = = a a exp i E t E 0 2 2 2 Electric susceptibility and permittivity (Phasor) ( ) + i 2 2 a 0 0 2 / n Q M ~ = a 0 2 2 0 ( ) / 2 0 n Q M ( ) ~ ~ = + = 0 1 1 + 2 i 2 14

  15. Refraction index and electric susceptibility Relation between refraction index and electric susceptibility / 0 n + = = + = + = : ' 1 " ' n the n n ~ ~ n = ~ ~ ~ ~ n 2 / 1 complex. be must 0 ~ ~ n Let ' i , " 2 n ' i " n n = + imaginary : " 2 real part n = 2 ' " part EXERCISE: Derive the above relation between n , n and , . 15

  16. Real and imaginary parts of n and Electric susceptibility ( 0) ( ) 2 0 i 2 + 2 / n Q M ~ = + = a 0 ' i " 2 ( ) 2 ( ) S + 2 0 2 0 2 / 2 S = 0 0 Real part : ' ( ) 2 0 ( ) 2 2 2 2 2 0 2 4 0 S + S 2 2 / 2 + n Q = = 0 a Imaginary part : " where , S ( ) ( ) 2 2 2 0 2 M 2 2 0 2 2 4 0 0 Refractive index (n 1, n 1) ( ) S + + / 4 1 + = + 0 0 0 Real part : ' 1 1 n ( S ) 4 + ( ) 2 2 2 2 4 MV 0 0 0 / 0 1 = 0 Imaginary part : " n ( ) ( ) 2 2 + 2 2 4 MV 0 0 0 0 16

  17. Graph of refractive index Refractive index (non-dimensional) n : real part n : imaginary part Angular frequency (rad/s) 17

  18. 4. Absorption and dispersion of light in material What are the real and imaginary parts of refractive index? Electric field (phasor rep.) ( ) ( ) / i exp , 0 = z k E z t Ex ~ n ~ ~ ~ E = exp i t z t 0 c Replace n by n +in ( ) = E z t Ex , + c ' ' i " n n ~ ~ i exp z t 0 " n n ~ = i exp exp E z t z 0 c c Propagating wave n traditional Spatial damping n absorption 18 refractive index

  19. Absorption of light Absorption: n describes damping of wave by dielectric. Vacuum Vacuum Dielectric Damping of electric field during D D n " exp D c Damping of light intensity I ( E2) 2 " c n ( ) = exp exp D AD D = exp d opt 2 " c n Absorption : coefficien t A 1 A c = Optical : depth d opt 2 " n 19

  20. Dispersion: separation of colors n is a function of . Refraction is different among colors. In most cases, 0 . n ( blue)>n ( red) Blue ray bends more deeply that red ray does. Snell' n law s : = sin n sin n 1 1 2 2 refracttiv : index e of dielectric 1, 2 2 , 1 angles : 1,2 of incidence refraction and dielectric in ( ' When n 1, ( 2 ) ) n ' , . blue red b r EXERCISE: Prove the above inequality. 20

  21. 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 : Real part : Imaginary part Atomic property Electric dipole moment of atom 0: Resonance frequency : Damping constant n = + 2 2 ' n " = 1 ' n n ( ) S / 2 0 S 0 ' 2 ' " " ( ) 2 + 2 0 / 2 + 0 " ( ) 2 2 0 21

  22. 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

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