FI 3221 ELECTROMAGNETIC INTERACTIONS IN MATTER

Transcription

1 FI 3 ELECTROMAGNETIC INTERACTIONS IN MATTER Alexande A. Iskanda Physics of Magnetism and Photonics CLASSICAL MODEL OF PERMITTIVITY Loentz Model Dude Model Alexande A. Iskanda Electomagnetic Inteactions in Matte

2 REFERENCES Main A.M. Fox : Section.. and.., M. Dessel and G. Gune : Section 6. and 5. Sulementay Jai Singh : Section.3 S.A. Maie : Section. Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 ˆˆ~ Nˆ ˆ~ i ( ) ~ () () () ~ () ~ () () () n n ˆ~ () () () ~ () ˆ i ~( ) ~ () () () n n Nˆ n () () () () () () n i Alexande A. Iskanda Electomagnetic Inteactions in Matte 5

3 LINEAR DIELECTRIC RESPONSE OF MATTER Hamonic oscillation model can be used to aoximately modelled the fequency deendence of suscetibility the Loentz model. Behaviou of bound electons in an electomagnetic field. Chages in a mateial ae teated as hamonic oscillatos. Otical oeties of insulatos ae detemined by bound chages, d m Fsing Fdaming F alied field dt Alexande A. Iskanda Electomagnetic Inteactions in Matte 6 LINEAR DIELECTRIC RESPONSE OF MATTER d d m m C ee dt dt Hamonic oscillation model can be used to aoximately modelled the fequency deendence of suscetibility the Loentz model. Conside a hamonic alied field : i E E t alied field e alied field + -e, m e Alexande A. Iskanda Electomagnetic Inteactions in Matte 7 3

4 ATOMIC POLARIZABILIT Y Solve fo the diole moment e fom the following d d C it e Ee dt dt m yields the following steady-state the solution e i E m o e E m i i t e Alexande A. Iskanda Electomagnetic Inteactions in Matte 8 ATOMIC POLARIZABILIT Y Recall the definition of atomic olaizability E hence e m i Alexande A. Iskanda Electomagnetic Inteactions in Matte 9 4

5 SUSCEPTIBILIT Y AND PERMITTIVIT Y Polaization is defined as the diole moment e unit volume P j je NE E V j V j N is the atomic density e unit volume. Thus, fom the evious esults, we obtain Ne N m i i Ne whee is defined as the lasma fequency. m Alexande A. Iskanda Electomagnetic Inteactions in Matte SUSCEPTIBILIT Y AND PERMITTIVIT Y Recall the elation between suscetibility and emittivity i i Alexande A. Iskanda Electomagnetic Inteactions in Matte 5

6 FREQUENCY DEPENDENCE OF PERMITTIVIT Y Fom the last elation of suscetibility and emittivity, we obtain Alexande A. Iskanda Electomagnetic Inteactions in Matte REFRACTIVE INDEX n n n n n, nn << : high n low v hase : stong deendence v hase, lage absotion ( n ) >> : n = v hase = c Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 6

7 KRAMERS KRONIG SUM RULE Fo the ange highe than the fequency of lagest absotion,, we can aoximate () On the othe hand, fom KK theoy ~ () () ~ ( ) () ( ) P d ~ ( ) d We can make the following identification ( ) d also n( ) ( ) d Alexande A. Iskanda Electomagnetic Inteactions in Matte 4 OPTICAL PROPERTIES OF METAL Electons in metal ae fee (fee electon gas model, Dude model), howeve in its motion thee ae collisions, hence its equation of motion is given as d d m m ee alied field dt dt The mean fee ath of the electon is chaacteized by its elaxation time t, hence we can wite dv v m m eealied field dt t i t Assume a time hamonic alied Ealied field E e field. Alexande A. Iskanda Electomagnetic Inteactions in Matte 6 7

8 OPTICAL PROPERTIES OF METAL We ae looking fo a solution in the fom of Substituting yields v e E m t i v v i t e Alexande A. Iskanda Electomagnetic Inteactions in Matte 7 CONDUCTIVIT Y Recall the cuent density in metal is given in tems of the dift velocity, the chage and its volume density, hence e N J Nev E m t i Comaing with the Ohm s law J E, we deduce Ne m Ne t, t i it m Whee is called the DC conductivity. Alexande A. Iskanda Electomagnetic Inteactions in Matte 8 8

9 CONDUCTIVIT Y Seaating the eal and imaginay ats of the conductivity, yield t i i t t Ne Recall the definition of lasma fequency then m t t t Ne t t t t m t t Alexande A. Iskanda Electomagnetic Inteactions in Matte 9 CONDUCTIVIT Y Some values of Ne m ae Metal (ev) Al 5. Cu 8.8 Ag 9. Au 9. Alexande A. Iskanda Electomagnetic Inteactions in Matte 9

10 BOUND AND CONDUCTION ELECTRONS Recall that thee ae actually two kinds of electons in a metal, namely the bound electons and conduction electons. Both of these electons contibutes to the emittivity. Conside the Amee-Maxwell equation, D H J t with time-vaying electic field and the cuent density as follows i t E E e J E alied field Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 BOUND AND CONDUCTION ELECTRONS We have D H J i B E E t The effective emittivity consist of contibution fom the bound chages B () and the conduction electons D H J i B E t i i i B E i eff E eff B i Alexande A. Iskanda Electomagnetic Inteactions in Matte 4

11 DIELECTRIC CONSTANT OF METAL Using the exession of conductivity We have i.e. eff B i i i t t t i t t B t t B t t Alexande A. Iskanda Electomagnetic Inteactions in Matte 5 DIELECTRIC CONSTANT OF METAL At fequencies visible since visible t >>, then i it t t Hence, effective emittivity become eff B i B i 3 t t t B t t Alexande A. Iskanda Electomagnetic Inteactions in Matte 6

12 DIELECTRIC CONSTANT OF METAL Define Ne t m Then eff B i B i 3 t t can be witten as Bounds electons eff B i 3 t Fee electons Alexande A. Iskanda Electomagnetic Inteactions in Matte 7 EXAMPLE : ALUMINIUM eff B i 3 t At >>, simila behaviou as dielectic Alexande A. Iskanda Electomagnetic Inteactions in Matte 8

13 COMPARISON WITH EXPERIMENTAL DATA Dielectic function () of the fee electon gas (solid line) fitted to the liteatue values of the dielectic data fo gold [Johnson and Chisty, 97] (dots). Inteband tansitions limit the validity of this model at visible and highe fequencies. Alexande A. Iskanda Electomagnetic Inteactions in Matte 9 COMPARISON WITH EXPERIMENTAL DATA Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 3

14 CONTRIBUTION FROM INTERBAND TRANSITION The disceancies between the exeimental data and the theoetical model can be econcile by consideing inteband tansition of the electons. To this end, we add exta tems in the emittivity exession that coesond to this inteband tansition. ( ) i A e i e i Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 i i CONTRIBUTION FROM INTERBAND TRANSITION Alexande A. Iskanda Electomagnetic Inteactions in Matte 3 4

15 HOMEWORK The Clausius-Mossotti equation 3 V N elates the dielectic constant of a mateial to the olaisability of its atom. Deive this elationshi by caefully assuming that the field at oint in the dielectic can be witten as the sum of the local field (that consist of the extenal field and the field geneated by all othe molecule outside of the sheical exclusion) and the field induced by the diole in the sheical exclusion. Alexande A. Iskanda Electomagnetic Inteactions in Matte 33 HOMEWORK E total ( ) E local E induced diole E ext E induced diole Elocal Recall that the Polaization vecto is defined as N P ( atomic density)( atomic olaizability)( local field) E V local Alexande A. Iskanda Electomagnetic Inteactions in Matte 34 5

16 HOMEWORK The following gahs show the eal and imaginay at of the emittivity function of an unknown dielectic mateial Alexande A. Iskanda Electomagnetic Inteactions in Matte 35 HOMEWORK Fom the evious gahs, estimate the esonant fequency of the Loentz oscillato model. Estimate the lasma fequency of the model. With the electon mass value of m e = kg, and 8.85 C N m, estimate the valence electon density N that contibute to this Loentz oscillato model. Estimate the daming constant of the mateial. Alexande A. Iskanda Electomagnetic Inteactions in Matte 36 6