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Lorentz oscillator model
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<h2 id="derivation-of-electron-motion">Derivation of electron motion</h2> <p>The model is derived by modeling an electron orbiting a massive, stationary nucleus as a <a href="/facts/Mass-spring-damper_model/NAmKTWEB">spring-mass-damper system</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The electron is modeled to be connected to the nucleus via a hypothetical spring and its motion is damped by via a hypothetical damper. The damping force ensures that the oscillator's response is finite at its resonance frequency. For a time-harmonic driving force which originates from the electric field, <a href="/facts/Newton%27s_laws_of_motion/XKzGTXK1">Newton's second law</a> can be applied to the electron to obtain the motion of the electron and expressions for the <a href="/facts/Electric_dipole_moment/ou2LDUVg">dipole moment</a>, <a href="/facts/Polarization_density/2oQv9EHx">polarization</a>, <a href="/facts/Electric_dipole_moment/ou2LDUVg">susceptibility</a>, and <a href="/facts/Relative_permittivity/vnKbPTn1">dielectric function</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Equation of motion for electron oscillator: F net = F damping + F spring + F driving = m d 2 r d t 2 − m τ d r d t − k r − e E ( t ) = m d 2 r d t 2 d 2 r d t 2 + 1 τ d r d t + ω 0 2 r = − e m E ( t ) {\displaystyle {\begin{aligned}\mathbf {F} _{\text{net}}=\mathbf {F} _{\text{damping}}+\mathbf {F} _{\text{spring}}+\mathbf {F} _{\text{driving}}&=m{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\\[1ex]{\frac {-m}{\tau }}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}-k\mathbf {r} -{e}\mathbf {E} (t)&=m{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\\[1ex]{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}+{\frac {1}{\tau }}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}+\omega _{0}^{2}\mathbf {r} \;&=\;{\frac {-e}{m}}\mathbf {E} (t)\end{aligned}}} </p><p>where </p> <ul><li> r {\displaystyle \mathbf {r} } is the displacement of charge from the rest position,</li> <li> t {\displaystyle t} is time,</li> <li> τ {\displaystyle \tau } is the relaxation time/scattering time,</li> <li> k {\displaystyle k} is a constant factor characteristic of the spring,</li> <li> m {\displaystyle m} is the effective mass of the electron,</li> <li> ω 0 = k / m {\textstyle \omega _{0}={\sqrt {k/m}}} is the resonance frequency of the oscillator,</li> <li> e {\displaystyle e} is the elementary charge,</li> <li> E ( t ) {\displaystyle \mathbf {E} (t)} is the electric field.</li></ul> <p>For time-harmonic fields: E ( t ) = E 0 e − i ω t {\displaystyle \mathbf {E} (t)=\mathbf {E} _{0}e^{-i\omega t}} r ( t ) = r 0 e − i ω t {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}e^{-i\omega t}} </p><p>The stationary solution of this equation of motion is: r ( ω ) = − e m ω 0 2 − ω 2 − i ω / τ E ( ω ) {\displaystyle \mathbf {r} (\omega )={\frac {\frac {-e}{m}}{\omega _{0}^{2}-\omega ^{2}-i\omega /\tau }}\mathbf {E} (\omega )} </p><p>The fact that the above solution is <a href="/facts/Complex_number/2w7mqI7e">complex</a> means there is a time delay (phase shift) between the driving electric field and the response of the electron's motion.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="dipole-moment">Dipole moment</h2> <p>The displacement, r {\displaystyle \mathbf {r} } , induces a dipole moment, p {\displaystyle \mathbf {p} } , given by p ( ω ) = − e r ( ω ) = α ^ ( ω ) E ( ω ) . {\displaystyle \mathbf {p} (\omega )=-e\mathbf {r} (\omega )={\hat {\alpha }}(\omega )\mathbf {E} (\omega ).} </p><p> α ^ ( ω ) {\displaystyle {\hat {\alpha }}(\omega )} is the polarizability of single oscillator, given by α ^ ( ω ) = e 2 m 1 ( ω 0 2 − ω 2 ) − i ω / τ . {\displaystyle {\hat {\alpha }}(\omega )={\frac {e^{2}}{m}}{\frac {1}{(\omega _{0}^{2}-\omega ^{2})-i\omega /\tau }}.} </p><p>Three distinct <a href="/facts/Scattering/nMou7lEH">scattering</a> regimes can be interpreted corresponding to the dominant denominator term in the dipole moment:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <table><tbody><tr><th>Regime</th><th>Condition</th><th>Dispersion Scaling</th><th>Phase Shift</th></tr><tr><td><a href="/facts/Thomson_scattering/hUj6gSgB">Thomson scattering</a></td><td> ω 2 ≫ ω τ , ω 0 2 {\displaystyle \omega ^{2}\gg {\frac {\omega }{\tau }},\omega _{0}^{2}} </td><td> 1 {\displaystyle 1} </td><td>0°</td></tr><tr><td>Shneider-Miles scattering</td><td> ω τ ≫ | ω 0 2 − ω 2 | {\displaystyle {\frac {\omega }{\tau }}\gg |\omega _{0}^{2}-\omega ^{2}|} </td><td> ω 2 {\displaystyle \omega ^{2}} </td><td>90°</td></tr><tr><td><a href="/facts/Rayleigh_scattering/o0MQtPMM">Rayleigh scattering</a></td><td> ω 0 2 ≫ ω 2 , ω τ {\displaystyle \omega _{0}^{2}\gg \omega ^{2},{\frac {\omega }{\tau }}} </td><td> ω 4 {\displaystyle \omega ^{4}} </td><td>180°</td></tr></tbody></table> <h2 id="polarization">Polarization</h2> <p>The polarization P {\displaystyle \mathbf {P} } is the dipole moment per unit volume. For macroscopic material properties N is the density of charges (electrons) per unit volume. Considering that each electron is acting with the same dipole moment we have the polarization as below P = N p = N α ^ ( ω ) E ( ω ) . {\displaystyle \mathbf {P} =N\mathbf {p} =N{\hat {\alpha }}(\omega )\mathbf {E} (\omega ).} </p> <h2 id="electric-displacement">Electric displacement</h2> <p>The electric displacement D {\displaystyle \mathbf {D} } is related to the polarization density P {\displaystyle \mathbf {P} } by D = ε ^ E = E + 4 π P = ( 1 + 4 π N α ^ ) E {\displaystyle \mathbf {D} ={\hat {\varepsilon }}\mathbf {E} =\mathbf {E} +4\pi \mathbf {P} =(1+4\pi N{\hat {\alpha }})\mathbf {E} } </p> <h2 id="dielectric-function">Dielectric function</h2> <p>The complex dielectric function is given the following (in <a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>): ε ^ ( ω ) = 1 + 4 π N e 2 m 1 ( ω 0 2 − ω 2 ) − i ω / τ {\displaystyle {\hat {\varepsilon }}(\omega )=1+{\frac {4\pi Ne^{2}}{m}}{\frac {1}{(\omega _{0}^{2}-\omega ^{2})-i\omega /\tau }}} where 4 π N e 2 / m = ω p 2 {\displaystyle 4\pi Ne^{2}/m=\omega _{p}^{2}} and ω p {\displaystyle \omega _{p}} is the so-called <a href="/facts/Plasma_frequency/B7lKhaMT">plasma frequency</a>. </p><p>In practice, the model is commonly modified to account for multiple absorption mechanisms present in a medium. The modified version is given by<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> ε ^ ( ω ) = ε ∞ + ∑ j χ j L ( ω ; ω 0 , j ) {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+\sum _{j}\chi _{j}^{L}(\omega ;\omega _{0,j})} where χ j L ( ω ; ω 0 , j ) = s j ω 0 , j 2 − ω 2 − i Γ j ω {\displaystyle \chi _{j}^{L}(\omega ;\omega _{0,j})={\frac {s_{j}}{\omega _{0,j}^{2}-\omega ^{2}-i\Gamma _{j}\omega }}} and </p> <ul><li> ε ∞ {\displaystyle \varepsilon _{\infty }} is the value of the dielectric function at infinite frequency, which can be used as an adjustable parameter to account for high frequency absorption mechanisms,</li> <li> s j = ω p 2 f j {\displaystyle s_{j}=\omega _{p}^{2}f_{j}} and f j {\displaystyle f_{j}} is related to the strength of the j {\displaystyle j} th absorption mechanism,</li> <li> Γ j = 1 / τ {\displaystyle \Gamma _{j}=1/\tau } .</li></ul> <p>Separating the real and imaginary components, ε ^ ( ω ) = ε 1 ( ω ) + i ε 2 ( ω ) = [ ε ∞ + ∑ j s j ( ω 0 , j 2 − ω 2 ) ( ω 0 , j 2 − ω 2 ) 2 + ( Γ j ω ) 2 ] + i [ ∑ j s j ( Γ j ω ) ( ω 0 , j 2 − ω 2 ) 2 + ( Γ j ω ) 2 ] {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{1}(\omega )+i\varepsilon _{2}(\omega )=\left[\varepsilon _{\infty }+\sum _{j}{\frac {s_{j}(\omega _{0,j}^{2}-\omega ^{2})}{\left(\omega _{0,j}^{2}-\omega ^{2}\right)^{2}+\left(\Gamma _{j}\omega \right)^{2}}}\right]+i\left[\sum _{j}{\frac {s_{j}(\Gamma _{j}\omega )}{\left(\omega _{0,j}^{2}-\omega ^{2}\right)^{2}+\left(\Gamma _{j}\omega \right)^{2}}}\right]} </p> <h2 id="complex-conductivity">Complex conductivity</h2> <p>The complex <a href="/facts/Optical_conductivity/4KphJbyA">optical conductivity</a> in general is related to the complex dielectric function (in <a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>) as σ ^ ( ω ) = ω 4 π i ( ε ^ ( ω ) − 1 ) {\displaystyle {\hat {\sigma }}(\omega )={\frac {\omega }{4\pi i}}\left({\hat {\varepsilon }}(\omega )-1\right)} </p><p>Substituting the formula of ε ^ ( ω ) {\displaystyle {\hat {\varepsilon }}(\omega )} in the equation above we obtain σ ^ ( ω ) = N e 2 m ω ω / τ + i ( ω 0 2 − ω 2 ) {\displaystyle {\hat {\sigma }}(\omega )={\frac {Ne^{2}}{m}}{\frac {\omega }{\omega /\tau +i\left(\omega _{0}^{2}-\omega ^{2}\right)}}} </p><p>Separating the real and imaginary components, σ ^ ( ω ) = σ 1 ( ω ) + i σ 2 ( ω ) = N e 2 m ω 2 τ ( ω 0 2 − ω 2 ) 2 + ω 2 / τ 2 − i N e 2 m ( ω 0 2 − ω 2 ) ω ( ω 0 2 − ω 2 ) 2 + ω 2 / τ 2 {\displaystyle {\hat {\sigma }}(\omega )=\sigma _{1}(\omega )+i\sigma _{2}(\omega )={\frac {Ne^{2}}{m}}{\frac {\frac {\omega ^{2}}{\tau }}{\left(\omega _{0}^{2}-\omega ^{2}\right)^{2}+\omega ^{2}/\tau ^{2}}}-i{\frac {Ne^{2}}{m}}{\frac {\left(\omega _{0}^{2}-\omega ^{2}\right)\omega }{\left(\omega _{0}^{2}-\omega ^{2}\right)^{2}+\omega ^{2}/\tau ^{2}}}} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cauchy%27s_equation/KcM9cS7x">Cauchy equation</a></li> <li><a href="/facts/Sellmeier_equation/LJMlvtlA">Sellmeier equation</a></li> <li><a href="/facts/Forouhi%E2%80%93Bloomer_model/myoPDSmF">Forouhi–Bloomer model</a></li> <li><a href="/facts/Tauc%E2%80%93Lorentz_model/89ltnxlM">Tauc–Lorentz model</a></li> <li><a href="/facts/Brendel%E2%80%93Bormann_oscillator_model/3G9rVwMq">Brendel–Bormann oscillator model</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lorentz, Hendrik Antoon (1909). The theory of electrons and its applications to the phenomena of light and radiant heat. Vol. Bd. XXIX, Bd. 29. New York; Leipzig: B.G. Teubner. OCLC 535812. <a href="/wiki/OCLC_(identifier)" target="_blank">/wiki/OCLC_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dressel, Martin; Grüner, George (2002). "Semiconductors". Electrodynamics of Solids: Optical Properties of Electrons in Matter. Cambridge. pp. 136–172. doi:10.1017/CBO9780511606168.008. ISBN 9780521592536.{{cite book}}: CS1 maint: location missing publisher (link) <a href="9780521592536" target="_blank">9780521592536</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Dressel, Martin; Grüner, George (2002). "Semiconductors". Electrodynamics of Solids: Optical Properties of Electrons in Matter. Cambridge. pp. 136–172. doi:10.1017/CBO9780511606168.008. ISBN 9780521592536.{{cite book}}: CS1 maint: location missing publisher (link) <a href="9780521592536" target="_blank">9780521592536</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Almog, I. F.; Bradley, M. S.; Bulovic, V. (2011). "The Lorentz Oscillator and its Applications" (PDF). Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology. Retrieved 2021-11-24. <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_lorentz.pdf" target="_blank">https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_lorentz.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Colton, John (2020). "Lorentz Oscillator Model" (PDF). Brigham Young University, Department of Physics & Astronomy. Brigham Young University. Retrieved 2021-11-18. <a href="https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf" target="_blank">https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Colton, John (2020). "Lorentz Oscillator Model" (PDF). Brigham Young University, Department of Physics & Astronomy. Brigham Young University. Retrieved 2021-11-18. <a href="https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf" target="_blank">https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Colton, John (2020). "Lorentz Oscillator Model" (PDF). Brigham Young University, Department of Physics & Astronomy. Brigham Young University. Retrieved 2021-11-18. <a href="https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf" target="_blank">https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Patel, Adam (2021). "Thomson and collisional regimes of in-phase coherent microwave scattering off gaseous microplasmas". Scientific Reports. 11 (1): 23389. arXiv:2106.02457. Bibcode:2021NatSR..1123389P. doi:10.1038/s41598-021-02500-y. PMC 8642454. PMID 34862396. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8642454" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8642454</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Zhang, Z. M.; Lefever-Button, G.; Powell, F. R. (1998). "Infrared Refractive Index and Extinction Coefficient of Polyimide Films". International Journal of Thermophysics. 19 (3): 905–916. doi:10.1023/A:1022655309574. S2CID 116271335. Retrieved 2021-11-24. <a href="https://doi.org/10.1023/A:1022655309574" target="_blank">https://doi.org/10.1023/A:1022655309574</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>