Interaction of Atoms and Electromagnetic Waves

Transcription

1 Interaction of Atoms and Electromagnetic Waves Outline - Review: Polarization and Dipoles - Lorentz Oscillator Model of an Atom - Dielectric constant and Refractive index 1

2 True or False? 1. The dipole moment of this atom is p = q y, and points in the same direction as the polarizing electric field. y q q p E 2. The susceptibility relates the electric field to the polarization in this form: P = ɛ o χ e E NI 2πr 3. The refractive index can be written n = ɛo μ o ɛμ 2

3 Refractive Index: Waves in Materials Index of refraction How do we get from molecules/charges and fields to index of refraction? 3

4 Index of Refraction νλ = v p = c/n frequency wavelength When propagating in a material, c c/n λ λ o /n k k o /n Material n Vacuum 1 Air Water liquid Water ice 1.31 Diamond Silicon 3.96 at 5 x Hz λ (m) Radio waves IR UV Hard X-rays Microwaves Soft X-rays Gamma rays Visible E(t, z) =Re{E 0 e j(ωt k 0nz) } E(t, z) =Re{E 0 e j(ωt k z) } 4

5 Absorption Photograph by Hey Paul on Flickr. Why are these stained glass different colors? E(t, z) =Re{E 0 e αz/2 e j(ωt k 0nz) } Tomorrow: lump refractive index and absorption into a complex refractive index ñ Absorption coefficient Refractive index 5

6 Incident Solar Radiation How do we introduce propagation through a medium (atmosphere) into Maxwell s Equations? Image created by Robert A. Rohde / Global Warming Art. Used with permission. 6

7 Microscopic Description of Dielectric Constant E NO external field y E small amount of charge moved by field E Electron spring Nucleus Density of dipoles Electric field polarizes molecules P = Nδx equivalent to P = ɛ o χ e E 7

8 Lorentz Oscillator Lorentz was a late nineteenth century physicist, and quantum mechanics had not yet been discovered. However, he did understand the results of classical mechanics and electromagnetic theory. Therefore, he described the problem of atom-field interactions in these terms. Lorentz thought of an atom as a mass ( the nucleus ) connected to another smaller mass ( the electron ) by a spring. The spring would be set into motion by an electric field interacting with the charge of the electron. The field would either repel or attract the electron which would result in either compressing or stretching the spring. B x Electron spring Nucleus Hendrik Lorentz ( ) Nobel in 1902 for Zeeman Effect Lorentz was not positing the existence of a physical spring connecting the electron to an atom; however, he did postulate that the force binding the two could be described by Hooke's Law: where y is the displacement from equilibrium. If Lorentz's system comes into contact with an electric field, then the electron will simply be displaced from equilibrium. The oscillating electric field of the electromagnetic wave will set the electron into harmonic motion. The effect of the magnetic field can be omitted because it is miniscule compared to the electric field. 8 Image in public domain

9 Springs have a resonant frequency d 2 y Hooke s Law m = ky dt 2 d 2 y k = y dt 2 m k k Solution y(t) =Asin(t )+Bcos(t ) m m ω 0 ω 0 So we can write: ω 2 k o = k = mωo 2 m 9

10 Microscopic Description of Dielectric Constant B x Electron spring Nucleus Electron mass Restoring force (binding electron & nucleus) E field force Damping 10

11 Solution using complex variables Lets plug-in the expressions for and y into the differential equation from slide 9: Natural resonance d 2 d q y(t)+γ y(t)+ω y oy(t) 2 = E y (t) dt 2 dt m E y (t) =Re{ E y e jωt } y(t) =Re{ ye jωt } p +q q q ω 2 y + jωγy + ωoy 2 = m q 1 y = m (ωo 2 ω 2 )+jωγ E y E y 11

12 Oscillator Resonance Driven harmonic oscillator: Amplitude and Phase depend on frequency Low frequency At resonance High frequency medium amplitude large amplitude vanishing amplitude y Displacement, Displacement, Displacement and in phase with 90º out of phase with E y in antiphase E y y y E y 12

13 Polarization Since charge displacement, y, is directly related to polarization, P, of our material we can then rewrite the differential equation: D = ɛ o E + P P y = Nqy For linear polarization in ŷ direction d d ( + γ + ω 2 Nq 2 o)p y (t) = E y (t) =ɛ ω 2 2 dt m pe y (t dt ω 2 Nq 2 p = ɛo m P y (t) =Re{P y e jωt } o ) ω 2 p P y = ɛ y (ωo 2 oe ω 2 )+jγω 13

14 Dielectric Constant from the Lorentz Model D = ɛ o E + P ω 2 p P = ɛoe (ω 2 ω 2 )+jγω [o ] ωp D 2 = ɛ o 1+ (ω 2 o ω 2 )+jγω E ɛ = ɛ o [ ω 2 p 1+ (ω 2 o ω 2 )+jγω ] 14

15 Microscopic Lorentz Oscillator Model ω 2 p P = ɛ (ωo 2 oe ω 2 )+jγω ω 2 Nq 2 p = ɛo m ɛ = ɛ r jɛ i 15

16 Real and imaginary parts ω 2 p ɛ = (ω 2 o ω 2 )+jωγ = (ω 2 o p(ω o ω 2 ) ωpωγ 2 j ω 2 ) 2 + ω 2 γ 2 (ωo 2 ω 2 ) 2 + ω 2 γ 2 ω 2 2 ɛ = ɛ r jɛ i ωp 2 (ωo 2 ω 2 ) ωp(ω o ω ) ɛ r = ɛ i = (ωo 2 ω 2 ) 2 + ω 2 γ 2 (ωo 2 ω 2 ) 2 + ω 2 γ 2 16

17 Dielectric constant of water 80 Magnitude Frequency (GHz) Microwave ovens usually operate at 2.45 GHz 17

18 Complex Refractive Index 18

19 Absorption Coefficient 1 n = ɛ r + ɛ 2 r + ɛ 2 i 2 1 κ = ɛ r + ɛ 2 r + ɛ 2 i 2 E(t, z) =Re{E e αz/2 e j(ωt k 0nz) 0 } Absorption Refractive index 2π α =2k κ =2 κ [cm -1 o ] λ o 19

20 Absorption E(t, z) =Re{E 0e αz/2 e j(ωt k0nz) } Absorption coefficient Refractive index I(z) =I o e αz Beer-Lambert Law or Beer s Law 20

21 Key Takeaways Lorentz Oscillator Model Electron spring Nucleus d 2 y dy m = mω 2 dt 2 o y + qe y mγ dt ω 2 p P = ɛ o E ñ = n jκ (ω 2 o ω 2 )+jγω α =2k o κ ɛ = ɛ r jɛ i E(t, z) =Re{E e αz/2 e j(ωt 0 k 0 nz) } Decay Absorption coefficient Refractive index I(z) =I o e αz Beer s Law 21

22 MIT OpenCourseWare Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: