## Classical Lorentz model ### Model definition and solution Classical Lorentz model is introduced to answer how light interact with matters, namely **light-matter interaction**. The primary assumption is, the nucleus of the atom is much more massive than the electron, so it can be considered as a fixed wall. And electrons could be imagined as a finite mass connected to a spring, consequently bounded to the nuclei, the force of the spring is the binding force (of any kind). This is justified when the displacement is small enough. >[!Note]- >The assumptions are: >- nucleus are much more massive than electrons and can be considered with infinite mass. >- the binding force behaves like a spring regardless of the binding type, when the displacement is small enough. ![[Drawing 2024-03-17 00.49.40.excalidraw.svg]] This soon convert the problem to a forced, dissipative spring model. The equation of motion for the electron against the nuclei would be: $\begin{aligned} m \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} &=F_{driving} + F_{damping}+F_{spring}\\ m \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} &=-eE_{x} -\frac{m\gamma \mathrm{d}x}{\mathrm{d}t}-\kappa x \end{aligned}$ and $\gamma$ is damping rate, $\gamma =\frac{1}{\tau}$, $\tau$ is called relaxation time or mean free time. Eventually, we have $m \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} + m\gamma\frac{\mathrm{d}x}{\mathrm{d}t} + m\omega_{0}^{2}x = -eE_{x}$ with $\omega_{0}^{2}=\frac{\kappa}{m}$. >[!Note] >The $F_{driving}$ and $F_{spring}$ terms are easy to understand, namely the force of electric field and the force of the bounding nuclei. The term $F_{damping}$ is the damping coupled with the neighboring lattice (so dissipative). And for the spring, $\kappa$ is a material property. >The damping rate is given by relaxation time, $\tau$, which is defined as the time that the amplitude of the system decays to $\frac{1}{e}$. >The $\omega_{0}=\frac{\kappa}{m}$ is the intrinsic frequency (or called natural frequency), this is valid for any resonance system. See the wiki page [damping](https://en.wikipedia.org/wiki/Damping) for more information. With the driving field being harmonic, $E=E_{0} e^{i(kx-\omega t)}$, also assume the displacement is harmonic, $x = x_{0}e^{i(kx-\omega t)}$, we have $x_{0} = \frac{eE_{0}}{m[(\omega_{0}^{2} - \omega^{2} - i\gamma \omega)]}$. >[!Notice] >The $x_{0}$ here has a complex expression (due to the damping term), this is also the case for many of the later terms. So notice the expression of them. > >If above driving potential is not harmonic, we'll have nonlinear behavior. > >The harmonic wave, or say sinusoidal wave are widely available commercially, so people care about the the most. One can notice that for $\gamma \neq 0$, the proportionality factor between $x_{0}$ and $E_{0}$ would be complex, this means the displacement $x$ and the field $E$ does not, if damping exist, in phase. By rewriting $x_{0}$ and separate the amplitude and phase, we have the expression being $x_0 = \frac{eE_0}{m} A e^{i\phi}$ with the amplitude $A = \frac{1}{[(\omega_0^2 - \omega^2)^2 + \gamma^2 \omega^2]^{1/2}}$, and the phase $\phi = \tan^{-1} \frac{\gamma \omega}{\omega_0^2 - \omega^2}$. >[!Note] >This can be interpreted as, when $\omega = \omega_{0}$, the displacement could follow, otherwise it requires time for the displacement to follow the field. This can also be seen in the following figure. The $x$ axis is the frequency of the external field $E$. ![[Drawing 2024-03-17 01.21.53.excalidraw.svg]] ### Polarization and relative permittivity under Lorentz model With the motion of the electron solved, we can then calculate the [[Introduction to polarization|polarization]]. For single electron, the induced dipole moment is $p=-ex$, and with the electron density $N$ known (typically we can know), the macroscopic polarization density $P$ would be: $P = -Nex \rightarrow P = \frac{-Ne^2 E_0}{m\left[\omega^2 - \omega_0^2 + i\gamma\omega\right]}$ And since $P=\varepsilon_{0} \chi E_{0}$, and $\chi = \frac{\varepsilon}{\varepsilon_{0}} = \varepsilon_{r}-1$, we can write the polarization in term of permittivity. So we have $\frac{-Ne^2 E_0}{m\left[\omega^2 - \omega_0^2 + i\gamma\omega\right]} = \varepsilon_{0} \left( \frac{\varepsilon}{ \varepsilon_{0}} - 1 \right) E_{0}$ If we introduce the plasma frequency $\omega_{p}$ (This is field independent, see later note for explanation), $\omega_{p} = \frac{Ne^{2}}{m \varepsilon_{0}}$ >[!Note]- >The plasma frequency is derived from the collective motion under electric field. The expression could be derived from the lossless Drude model. But we just apply it here. Also considering it is not related to Lorentz model, it could be derived independently from Newton's 2nd law. We just use the result here. >See [[#Special case of metals]], $\omega_{p}$ separates two regions. we can write the ratio of relative permittivity in $\omega$, $\omega_{0}$, and $\omega_{p}$, namely $\varepsilon_{r}(\omega) = \frac{\varepsilon(\omega)}{\varepsilon_0} = 1 + \frac{\omega_p^2}{(\omega_0^2 - \omega^2) - i\gamma\omega}$ and separate it into the real and imaginary parts by $\varepsilon_r(\omega) = \varepsilon_r'(\omega) + i\varepsilon_r''(\omega)$, we have $\varepsilon_r'(\omega) = 1 + \frac{\omega_p^2(\omega_0^2 - \omega^2)}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}$ $\varepsilon_r''(\omega) = -\frac{\gamma\omega\omega_p^2}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}$ With the material properties known, it is possible to plot how $\varepsilon_{r}$ changes with frequency. The figure is shown below. ![[Drawing 2024-03-17 01.53.49.excalidraw.svg]] For the imaginary part $\varepsilon_{r}''$: - Lorentzian shape and peaks at $\omega_{0}$. - Electromagnetic energy get absorbed. - FWHM of $\varepsilon_{r}''$ is given by damping $\gamma$. For the real part $\varepsilon_{r}'$: - Inflection point at $\omega_{0}$ and can take **negative values** in the region where $\omega_{0}<\omega<\omega_{p}$. - Negative sign of $\varepsilon_{r}'$ has very important consequence in photonics (for metal) >[!Note]- >I'll say a negative $\varepsilon'$ being a negative value is kind of counterintuitive, since we typically talk about this quantity in insulators. For metal it can be considered as reflectioxn. More complex cases will be specifically discussed when it comes to meta materials later. >>[!Note] >>Actually the sign of the real part of permittivity $\varepsilon'$ is related to how material response to the external field. Since $\mathbf{D} = \varepsilon_{0} \mathbf{E}+ \mathbf{P} = \varepsilon \mathbf{E}$, it is just the material (or say polarization) has the counter-interaction compared to the incident field. And in macroscopic picture it behaves as reflection for negative $\varepsilon$, while for positive it's reflection. The real and imaginary part of $\varepsilon_{r}$ are interdependent, and they are connected by the [[Kramers–Kronig relations]]. $\begin{aligned} \varepsilon_r'(\omega) &= 1+\frac{2}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{\omega'\varepsilon_r''(\omega')}{(\omega')^{2} - \omega^{2}} \mathrm{d}\omega'\\ \varepsilon''_{r}(\omega) &= -\frac{2\omega}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{\varepsilon_{r}'(\omega')}{(\omega')^{2} - \omega^{2}} \mathrm{d}\omega' \end{aligned} $ $\mathcal{P}$ is [Cauchy principal value](https://en.wikipedia.org/wiki/Cauchy_principal_value). This relation enables us to obtain the real part $\varepsilon_{r}'$ by measuring the imaginary part $\varepsilon''_{r}$, since the shape of the $\varepsilon_{r}'$ depends on the area under the $\varepsilon''_{r}$ curve, which means the absorption *causes* anomalous dispersion. (See figure above) ### Light-matter interaction for materials - For semiconductor, the $\varepsilon_{r}$ looks like the figure above. The model includes all terms in Lorentz model. > [!Example]- > **Special case: X-ray interaction and refractive index** > For X-rays (photon energy from hundreds of eV to hundreds of keV), the incident frequency $\omega$ is much greater than any material resonance frequency $\omega_0$: > $ > \omega \gg \omega_0 \quad \Rightarrow \quad \omega_0^2 - \omega^2 \approx -\omega^2 > $ > Thus, the Lorentz model simplifies the relative permittivity to: > $ > \varepsilon_r(\omega) \approx 1 - \frac{\omega_p^2}{\omega^2} + i \frac{\gamma \omega_p^2}{\omega^3} > $ > The corresponding refractive index is: > $ > n(\omega) \approx 1 - \delta + i\beta > $ > where > $ > \delta = \frac{\omega_p^2}{2\omega^2}, \quad \beta \propto \frac{\gamma \omega_p^2}{2\omega^3} > $ > Physically: > - $\delta$ causes slight phase delay, enabling total external reflection at grazing incidence. > - $\beta$ represents weak absorption. > > Therefore, in X-ray optics, conventional reflection vanishes except at shallow angles, and materials appear nearly transparent. - For metal, electrons are not bounded, so the spring term does not exist and $\omega_{0} = 0$. The model becomes Drude model. $\frac{\varepsilon}{\varepsilon_{0}} = \varepsilon_{r}(\omega) = 1- \frac{\omega_{p}^{2}}{\omega^{2} + i \gamma \omega}$ ![[Drawing 2024-03-17 17.51.33.excalidraw.svg]] - For plasma (or say lossless Drude model), there is no electron-phonon damping and spring term, so the model would be $\frac{\varepsilon}{\varepsilon_{0}} = \varepsilon_{r}(\omega) = 1- \frac{\omega_{p}^{2}}{\omega^{2}}$ ![[Drawing 2024-03-17 17.55.33.excalidraw.svg]] And the ratio of $\frac{\varepsilon''}{\varepsilon_{r}}$, from 0 to $\infty$, meaning materials being lossless to perfect conductor (no propagation allowed). ^b7b0c1 >[!Note] >This is also true when $\omega \gg \text{collision rate}$. Here collision rate $\propto \gamma$, collision time $\propto \frac{1}{\gamma}$. #### Special case of metals As mentioned before, for metals we have negative permittivity. Before $\omega_{p}$ we have reflection regime, and after we have transmission regime. At reflection regime, the displacement (polarization) is out of phase, the induced dipole has a different direction against the field, so a screening is created and no transmission allowed. If the frequency increase, the electron will not be able to follow the excitation, then the screening becomes no longer possible. Therefore, we have transmission. ![[Drawing 2024-03-17 18.13.29.excalidraw.svg]] For aluminum, the Drude model is good enough to describe, but for silver or gold, $\varepsilon_{r}$ is not accurate for high frequency (photon energy). This is due to intraband transition (namely, between levels within the same band). Such intraband transition make metals like gold, silver and copper having different color than the one at the plasma frequency. An additional term should be added to compensate this effect, so for metal, the $\varepsilon_{r}$ would be $\epsilon_r(\omega) = 1 - \frac{\omega_{pe}^2}{\omega^2 + i\gamma_e\omega} + \sum_j \frac{\omega_{pj}^2}{\omega_{0j}^2 - \omega^2 - i\gamma_j\omega} $ The first term is free electron term in Durde mode, while the second term includes the bound electron effect. >[!Notice] >This method provides possibility to include multiple $\omega_{0}$ in Lorentz model. Although still remaining classical and phenomenological, it is much better than the model proposed before, with only one $\omega_{0}$ and cannot describe multiple resonances. >>[!Note] >>The resonance frequency $\omega_{0}$ essentially corresponds to transition between energy levels. Namely $(E_{n}-E_{n-1})/\hbar$. Single frequency for sure cannot accurately describe the full picture. >[!Info] >See more at https://en.wikipedia.org/wiki/Lorentz_oscillator_model.