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Drude Dielectric Function (Permittivity)


The Drude model is often combined with other oscillator models to reproduce the observed permittivity of metals.

Stories to Tell

 * Relationship between optical conductivity, permittivity and refractive index.
 * Where does the formula come from?
 * Decomposition of optical properties of metals into interband and intraband components.

Physical Interpretation
The relaxation time $$\tau$$ and it's inverse, the scattering rate $$\gamma$$ are dominated by electron-phonon scattering in good metals like gold, silver, aluminum, lithium. For such metals, the scattering rate is well approximated from the temperature dependent DC resistivity:

$$1/\tau(T)=\gamma(T)=\epsilon_0\omega_p^2\rho_{DC}(T)$$

also predict the current as a response to a time-dependent electric field with an angular frequency $ω$. The complex conductivity is
 * $$\sigma(\omega) = \frac{\sigma_0}{1 - i\omega\tau}= \frac{\sigma_0}{1 + \omega^2\tau^2}+ i\omega\tau\frac{\sigma_0}{1 + \omega^2\tau^2}.$$

Errors in Time and Reliability
= Errors in Time = The shape parameter is $$\beta$$ and the scale parameter is $$\alpha$$. $$T$$ is the time at which we want to calculate the reliability, $$R = 1-\textrm{CFR}$$:
 * $$\textrm{CFR} = 1-\exp\left(-\left(\frac{T}{\alpha}\right)^{\beta}\right)$$


 * $$\ln(R) = -\left(\frac{T}{\alpha}\right)^{\beta}$$
 * $$\ln(-\ln(R)) = \beta\ln\left(\frac{T}{\alpha}\right)$$
 * $$\ln(-\ln(R)) = \beta\ln\left(\frac{T}{\alpha}\right)$$
 * $$\ln(-\ln(R)) = \beta\left(\ln(T) - \ln(\alpha)\right)$$

Let $$u=\ln(T) $$
 * $$u = \frac{1}{\beta} \ln(-\ln(R)) + \ln(\alpha)$$

In the following, $$\hat{u}$$ is the expected value of $$u$$ from a fit and $$K_{\alpha_\mathrm{CI}}=z$$ is the number of std deviations away from the norm that we will calculate the CI for. In Python, z = -scipy.stats.norm.ppf((1 - CI) / 2). The Confidence bounds are estimated by:
 * $$u_\mathrm{U} = \hat{u} +

K_{\alpha_\mathrm{CI}} \sqrt{ \mathrm{Var} \left( \hat{u} \right) } $$
 * $$u_\mathrm{L} = \hat{u} -

K_{\alpha_\mathrm{CI}} \sqrt{ \mathrm{Var} \left( \hat{u} \right) } $$



\mathrm{Var} \left( \hat{u} \right) = \frac{ \mathrm{Var}(\hat{\alpha}) }{     \hat{\alpha^2} } + \frac{ \mathrm{Var}(\hat{\beta}) }   {        \hat{\beta^4} }    \left( \ln(-\ln(R)) \right)^2 -2 \left(        \frac{            \mathrm{Cov}(\hat{\beta},\hat{\alpha})        }{          \hat{\alpha} \hat{\beta^2}        }    \right) \ln(-\ln(R)) $$