# Optical constants of noble metals: the Drude model for microwave modelling

I have a question regarding the optical constants of noble metals.

According to Johnson and Christy's paper Optical Constants of Noble Metals (Phys. Rev. B 6, 4370–4379 (1972), doi:10.1103/PhysRevB.6.4370), in the Drude free electron theory the equation for the dielectric permittivity is $$\epsilon(\omega)=1-\frac{\omega_p^2}{\omega(\omega+i/\tau)}=1-\frac{\omega_p^2}{\omega(\omega+i\gamma)},$$ where $\epsilon(\omega)$ is the dielectric constant at frequency $\omega$, $\omega_p$ the plasma frequency, and $\gamma=1/\tau$ the collision frequency.

In the nanohub photonics database the constants can be calculated from different theoretical and experimental models.

However, in CST Microwave Studio I need to give the epsilon infinity value, which really comes from modified drude model: $$ε=ε(\infty)-\frac{ω_p^2}{ω^2+iγω}.$$ How do I get the value of $\epsilon(\infty)$ for gold? Is there any reference by using which we can directly convert the free electron theory Drude model to the modified Drude model?

I am specifically searching for the values of $ε(\infty)$, $ω_p$, and $γ$ for gold in the 700nm to 1100 nm range (specifically 830nm).

• Welcome, you can use latex by putting formula between two $signs. Also I have added a link to the article you mentioned. Commented Dec 16, 2012 at 21:21 • Perhaps the data at wave-scattering.com/drudefit.html will help you. Commented Dec 22, 2013 at 19:22 ## 1 Answer Unless I'm misinterpreting your question I think you're confused. The values do not change much with wavelength. Instead these values are supposed to provide a model for which the permittivity can be found over a range of wavelengths. So to answer your last question, the values that were posted by user21414 are correct for the range listed, as they are actually correct for the range 250-2000nm. However these values vary based on certain authors that have produced them e.t. Blaman, Zeman, Grady . Point is, there are many values that actually are accepted. The term you are looking for$\epsilon_ib(\omega)$is the value for the bound electrons within a material. This value must be accounted for in the drude model. Since most metals are mostly free electrons the value$\epsilon_ib(\omega)$, is a constant known as$\epsilon(\infty)$. This is the value that you were wondering about. The equation for it is: $$\epsilon_ib(\omega)=1+{\omega_1^2}/{(\omega_0^2-\omega^2-i\gamma\omega)}$$ This should give you the value for$\epsilon(\infty)\$ for all metals.

Also, if you had thoroughly read the Johnson & Christy paper you would know that the first model (first equation you listed) only relates to the second in that, the first model is inaccurate for real world applications.

To actually experimentally find these values you have to go back into Johnson & Christy's paper or one of the paper's to see how they experimentally achieved these values. However I don't think this was your question. I think your question was more concerned more how to find these values in tables and secondly where these values were derived from.

Also if you have access, I'd suggest getting a copy of Optical Metamaterials: Fundamentals and Applications by Wenshan Cai & Vladimir Shalaev, the second chapter of the book does an excellent job of describing this.

• Ok, +1 now you have enough reputation to post the link. Commented Nov 15, 2013 at 3:23