Question

5. [10 points] With a simplified classical model (electron-ion connected with a spring --> forced damping oscillator) for the behavior of electrons one can describe the frequency dependence of the permittivity (€.(w) = {(W)/€) in dielectrics. Show the full derivation procedure of (w) starting from the classical model.

Answer 1

twee plu) puust Then canotton becomes, From polarisation of above oscillation dpcas) ejust - asa P ( w) e just YCty = - Plt) dt? dt I driving & plups escut to we cay event derivative of exponential teems C Plue) independent of time) (just plues ejust t & jatue Blusel Taking dez W + dza to all the definition of In general for a to forced dans ped dy. Na + mr day towary dy dute Now can come to question, Lorentz oscillator model. electron spring like force. dp ma Amway= Fariving dez - Cone Eag evert علم mad f driving The exponentials cancers, and by solving for complex de2 Na Polonization obtain: coe Co were up = - Na Fairving Poto) = Eog. inucleus. (we-w²) + juw in the NO. tein comes substituting and expression for Fanning :- uloop Nq2 salid to the intana from calision The damping the electros. A and radiation emitted by there complete the Lorent's O Scalator equatros is determine the driving force In the case of a solid placed in electric field varying in with but is reft for Ey ale complete the model. Recall our equation for the polanzaton to are we the differential Nq² d² Plast dt dp (wet cez plast)= Eylwest), angular frequency + independent y direction of the displacement cohere is the the plasma recall and definition for polarization, B- no Xe concept of introduced You have been dielectric relative permittivity, also called constant le = 1+ Xe we can rewrite the above 0 definition of polarization as: P = cao ( 4o - 1) accomodate make slight modification comples result. instead of a real & Let there complex e such that. Pewn= cool la - 1) Foy combining with resulti- 1+ a we = 1+ (202) tjuen caehw²g+juer to write eauation we can the will frequency with electric field varies. substituting frequency in the followin following way. pc cost) = Co wa Ey Cuest), Egon. By ewit) = Ery cosult a Relly (wt)} = Relay eroty OCY F driving be a -a ky d²Peart) dplaat dt cer learned since, plasma frequensy, Previous op²- Na² bave two different, but equivalent formulas for the polarization: i B = NZ - Nal-}} = - Nağı 2. P = no Xe B where to Electric succeptibility 1 & Gwluo Plne,t) = plus costust) - Rel Place) eiusty