Plasma and helicon wave frequencies in iron according to Drude model

Question

The first chapter of Ashcroft and Mermin's Solid State Physics discusses electromagnetic waves in metals. One of the exercises requires calculation of the plasma and helicon frequenies using the Drude model. For iron, I got $2.3*10^{16}$ for the plasma frequency and 11.5 for helicon wave frequency (radians per second). The first seems too high and the second seems too low. Are they reasonable?

The given formula for the plasma frequency is $\omega_p^2=\frac{4\pi n e^2}{m}$ where n is the density of charge carriers, e is the electron charge and m is the electron mass. The authors seem to be using gaussian units although this is not entirely clear. For n I used $1.7*10^{23}\:cm^{-3}$

My cyclotron frequency for a magnetic field of 10 kilogauss, $1.76*10^{11}$, seems to be right.

For the helicon wave frequency the given formula is $\omega=\omega_c(\frac{k^2c^2}{\omega_p^2})$ where $\omega_c$ is the cyclotron frequency and $\omega_p$ is the plasma frequency. The value of $k=2\pi/\lambda$ correesponds to wavelength of 1 cm.

Answer 1

For the plasma frequency, you're only about one order of magnitude off; the correct value is $9.89 \times 10^{14}$ Hz (source). The reason for the difference is that because of iron's long-range periodic crystal structure, electrons in iron have an effective mass significantly higher than their physical mass. Note that there is no single effective electron mass for a given substance, so you can't just go look up the effective mass in a table and substitute it into your equation; rather, the plasma frequency generally has to be determined by experiment.

For a metal, it is reasonable to expect helicon frequencies in the range of perhaps $10^1$ to $10^4$ Hz (for example, see this lab). Your value is definitely on the low side and probably isn't realistic. The error here is due to at least two things, the incorrect plasma frequency and not using the cyclotron effective mass to compute the cyclotron frequency.

In any case, I should add that you've applied the Drude model correctly; the fact that the results do not match what we observe in real life is due to limitations of the model, not any mistake on your part.