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

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

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

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)

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.

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

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.

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

Relation between electric dipole moment of atom and electric polarization

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

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”

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

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

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)

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

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

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

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

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)

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.

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

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