Dive into Electromagnetic Response and Dielectric Materials

phy 712 electrodynamics 11 11 50 am in olin 103 n.w
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Explore the concepts of real and imaginary contributions to electromagnetic response, frequency dependence of dielectric materials, the Drude model, and Kramers-Kronig relationships. Understand the vibration of charged particles near equilibrium and the implications for energy dissipation. Delve into induced dipoles, displacement fields, and the Drude model's expression for permittivity and dielectric function.

  • Electromagnetic response
  • Dielectric materials
  • Drude model
  • Frequency dependence
  • Induced dipoles
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  1. PHY 712 Electrodynamics 11-11:50 AM in Olin 103 Notes for Lecture 18: Continue reading Chapter 7 1. Real and imaginary contributions to electromagnetic response 2. Frequency dependence of dielectric materials; Drude model 3. Kramers-Kronig relationships 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 1

  2. Choose project topic by Fri. 3/4/2022 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 2

  3. 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 3

  4. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 Note that: > 0 represents dissipation of energy. 0 represents the natural frequency of the vibration; 0=0 would represent a free (unbound) particle 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 4

  5. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 E 1 q = i t r r r 0 For , e 0 0 2 0 2 m i Induced dipole : 2 E 1 q = = i t p r 0 q e 2 0 2 m i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 5

  6. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 Displacement field: = D P = = p + E E P r 0 ( ) 3 r p N f i i i i i i number of dipoles/volume fraction of type dipoles N f i i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 6

  7. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png Drude model expression for permittivity: = = + D E E = + P E p N f 0 0 i i i 2 E 1 q i t q = = p r 0 i e i i 2 i 2 m i i i 2 1 q m i t = + E E 1 i e N f 0 0 i 2 i 2 i i 0 i i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 7

  8. Drude model dielectric function: ( ) 0 2 1 q i = + 1 i N f i 2 2 m i 0 i i i ( ) 0 ( ) 0 = + R I i ( ) 0 2 2 2 q i i = + 1 i R N f ( ) i 2 m 2 + 2 2 2 0 i i i ( ) 0 2 q i = i i I N f ( ) i 2 m 2 + 2 2 2 0 i i i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 8

  9. Drude model dielectric function: ( ) R 0 ( ) I 0 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 9

  10. Drude model dielectric function some analytic properties: ( ) 2 1 q 1 = + i N f i i 2 i 2 m i 0 0 i i ( ) 2 1 q 1 For i N f i i 2 m i 0 0 i 2 P 2 P 1 2 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 10

  11. Drude model dielectric function some analytic properties: ( ) 0 0 particle free a ing (represent 0 For = 2 1 q 1 = + i N f i i 2 i 2 m i i i of charge mass , q m 0 0 0 ( ) 0 2 2 1 1 q q 1 = + + 0 m i N f iNf ( ) 0 i i i 2 i 2 m 0 i 0 0 0 0 i i ( ) 0 ( ) 0 + b i Some details: = D E = J E + J b D E i ( ) = = = = + H E E i i b b t t 2 1 q m ( ) = 0 Nf ( ) 0 i 0 0 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 11

  12. Analytic properties of the dielectric function (in the Drude model or from first principles -- Kramers-Kronig transform Consider Cauchy's integral formula for an analytic function ( ): f z 1 i f(z) z - ( ) = = ( ) dz f z 0 f dz 2 includes Im(z) Re(z) 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 12

  13. Kramers-Kronig transform -- continued Im(z) Re(z) =0 1 i 1 i f(z) z - f(z ) z - f(z) z - ( ) = = + R f dz dz dz R 2 2 R includes rest 1 1 1 f(z ) f(z ) ( ) = = + ( ) R R f dz P dz f R R 2 2 2 z - z - i i R R 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 13

  14. Kramers-Kronig transform -- continued 1 1 f(z ) ( ) = + ( ) R f P dz f R 2 2 z - i R ( ) z R ( ) z R ( ) z R = + Suppose : f f if R I ( ) z R ( ) z R + - R 1 1 f if ( ( ) ( ) ) + = R I f if P dz R I R 2 2 z i 1 f (z ) ( ) = I R f P dz R R z - R 1 f (z ) ( ) = R R f P dz I R z - R 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 14

  15. Kramers-Kronig transform -- continued 1 f (z ) ( ) = I R f P dz R R z - R 1 f (z ) ( ) = R R f P dz I R z - R This Kramers-Kronig transform is useful for the dielectric function ( ) ( ) when 1 f z R 0 ( ) ( ) f z Must show that: 1. is analytic for 0 f z z I 2. vanishes for z 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 15

  16. u a us b Some practical considerat ions Principal integratio parts n : u b b lim s a a + s = + ( ) ( ) ( ) P du g u du g u du g u 0 u Example : u b b lim 1 1 1 s a a + s = + P du du du 0 u-u u-u u-u s s s u lim b u b u = + = ln ln ln s s 0 u a u a s s 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 16

  17. More practical considerat ions ( ) dielectric For function : ( ) ( ) R = = * ( ( ) ) ( ) ( ) I R = I Analytic properties the dielectric function which justify ( ) z ( ) f z the treatment of 1 0 ( ) ( ) f z Must show that: 1. is analytic for 0 f z z I 2. vanishes for (for 0) z z I 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 17

  18. Analysis for Drude model dielectric function: ( ) 0 2 1 2 q 1 = + i N f i 2 m i i 0 i i i ( ) 0 2 1 2 q z ( ) z = 1 = Let i f N f i 2 m z iz i 0 i i i For z i 2 1 q ( ) z vanishes large at i f N f z i 2 z m i 0 i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 18

  19. Analysis for Drude model dielectric function continued -- Analytic properties: ( ) z 2 1 q m ( ) f z = = 1 i N f i 2 i 2 z iz i 0 0 i i ( ) f z = 2 i 2 has poles at 0 z z iz P P P i 2 = 2 i i i z i P 2 2 is analytic for ( ) ( ) f z ( ) Note that 0 0 z z P P 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 19

  20. ( ) 2 1 2 q z ( ) z i = 1 = i f N f i 2 i m z iz 0 0 z i i ( ) z 2 = 2 i has poles at 0 f z iz P P P i 2 = 2 i i i z i P 2 2 ( ) z ) ( ) z ( ) z Note that ( 0 analytic is for 0 f P P pz ( ) analytic f z ( ) pz 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 20

  21. Because of these analytic properties, Cauchys integral theorem results in: Kramers-Kronig transform for dielectric function: Note that these results will be useful for HW #17. ( ) 0 ( ) 0 1 ' 1 = d 1 ' R I P - ' ( ) with ( ) ; 1 ' 1 = d ' 1 I R P - = ' 0 0 ( ) ( ) ( ) ( ) = R R I I 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 21

  22. Further comments on analytic behavior of dielectric function "Causal" relationship between and fields: = + E D ( ) ( ) ( ) ( ) d G D r E r E r , , , t t t 0 0 ( ) ( ) e = d G i 1 0 0 Some details: Consider a convolution integral such as = ( ) ( ') ( g t h t ') ' where the functions ( ), ( ), and ( ) f t t dt f t g t h t are all well-defined functions with Fourier transforms such as 1 = = ' i t i t ( ) ( ') f t e ' ( ) ( ) f dt f t f e d 2 = ( ) ( It follows that: ( f ) ) g h 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 22

  23. Further comments on analytic behavior of dielectric function E D "Causal" relationship between and fields: = + ( ) ( ) ( ) ( ) d G D r E r E r , , , t t t 0 0 ( ) ( ) 1 ( ) = = d G i i ( ) 1 ( G ) = 1 G e d e 2 ( ) 0 0 0 2 1 q m N = For 1 f i i 2 i 2 i i 0 0 i i ( ) sin 2 q m N = / 2 i ( ) ( ) G f e i i i i 0 i i 2 i 2 i where / 4 i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 23

  24. Some details ( ) 1 1 = i iz ( ) 1 = ( ) f z e G e d dz 2 2 0 ( ) z 2 1 q m ( ) f z = = Let 1 i N f i = 2 i 2 z iz i 0 z 0 i i ( ) f z 2 i 2 has poles at 0 z iz P P P i 2 2 = = 2 i 2 i or i i i i z i z i P P 2 2 2 2 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 24

  25. 1 = = iz ( ) ( ) f z e Res( ) G dz i z P 2 P = iz z iz Note that: e e e R I Valid contour for ( ) 0 for G = 0 0 Valid contour for ( ) G = 0 zP ( ) sin 2 q m N /2 i i f e i i i 0 i i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 25

  26. 1 = = iz ( ) ( ) f z e Res( ) G dz i z P 2 P 2 i m ( ) z 1 q ( ) f z = = Let 1 N f i = 2 i 2 z iz i 0 z 0 i i ( ) f z 2 i 2 has poles at 0 z iz P P P i 2 2 = = 2 i 2 i or i i i i z i z i P P 2 2 2 2 ( ) sin 2 q m N = / 2 i ( ) ( ) G f e i i i i 0 i i 2 i 2 i 2 i 2 i where / 4 assuming / 4 0 i 02/28/2022 PHY 712 Spring 2022 -- Lecture 18 26

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