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