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

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

3. 1. Review of electromagnetic wave: Electromagnetic waves in vacuum VS
1. Review of electromagnetic wave: Electromagnetic waves in vacuum VS. dielectric material
In vacuum: ε0, μ0
In dielectric material: ε(≠ε0), μ0

4. 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)

5. Phasor representation
Waves can be represented by complex exponential function instead of real trigonometric function.
Real trigonometric function
Complex exponential function (Phasor rep.)
EXERCISE:
NOTE: “~” denotes phasor representation, and therefore it is complex.

6. 2. Electric dipole interaction: Force and potential energy
Force acting on charge and potential energy of electric dipole moment
Force acting on charge
Electric dipole moment and Polarization
Electric dipole moment
Electric
polarization
Potential energy

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.

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

9. Relation between electric dipole moment of atom and electric polarization

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

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”

12. Oscillatory motion of electron cloud
Motion of the center of the electron cloud ⇒ Damped harmonic oscillation
Equation of motion
Set z=0, φ0=0 for simplification
Phasor representation

13. Solution of damped oscillation
Equation of motion (phasor rep.)
Solution (phasor rep.)
EXERCISE: Solve the above differential equation.

14. 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)

15. Refraction index and electric susceptibility
Relation between refraction index and electric susceptibility
EXERCISE: Derive the above relation between n’, n” and χ’, χ”.

16. Real and imaginary parts of n and χ
Electric susceptibility (Γ≪ω0)
Refractive index (n’≃1, n”≪1)

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

18. 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”
Propagating wave
n’ ⇒traditional
refractive index
Spatial damping
n” ⇒absorption

19. Absorption of light
Absorption: n” describes damping of wave by dielectric.
Vacuum
Dielectric
Vacuum
Damping of electric field
during D D
Damping of light
intensity I (∝E2)

20. 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.
EXERCISE:
Prove the above inequality.

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
χ’ : Real part
χ” : Imaginary part
Atomic property
Electric dipole moment
of atom
ω0: Resonance
frequency
Γ : Damping constant

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