Engineering & Science
Drude Free-Electron Metal Conductivity & UV Plasma Frequency Formula
Calculate DC electrical conductivity sigma = (n * e^2 * tau) / m_e in MS/m and plasmon energy hbar * omega_p (in eV) explaining both metal Ohm's Law and why metals turn transparent in the deep ultraviolet!
Connect femtosecond electron scattering tau to copper conductivity & UV transparency!
Paul Drude (1900) modeled conduction electrons of number density n (m^-3), charge -e, and mass m_e accelerating under electric field E for an average collision relaxation time tau (in femtoseconds, 10^-15 s) before scattering off phonons or impurities—deriving microscopic Ohm's Law conductivity: sigma = (n * e^2 * tau) / m_e! For monovalent copper (n = 8.47e28 electrons/m^3) with room-temperature scattering time tau = 25.0 fs, Drude predicts sigma = 59.67 MS/m (5.967e7 S/m, matching experimental annealed copper 59.6 MS/m to 0.1%!)!
At optical frequencies, the free-electron gas oscillates collectively at the bulk Plasma Frequency omega_p = SQRT((n * e^2) / (eps0 * m_e)), corresponding to plasmon photon energy E_p = hbar * omega_p (in eV)! Below E_p (visible & infrared light), permittivity is negative and metals act as shiny mirrors; ABOVE E_p (10.81 eV in Cu, 8.99 eV in Ag, ~5.9 eV in alkali sodium/potassium), electrons cannot keep up with the wave and solid alkali metals suddenly turn optically TRANSPARENT in the ultraviolet!
| Noble Metal Free-Electron Gas | Carrier Density n & Scattering tau | Drude Conductivity sigma (MS/m) | Bulk Plasmon Energy Ep (eV) |
|---|---|---|---|
| Copper (Cu, 300 K) | n = 8.47e28 m^-3, tau = 25 fs | 59.67 MS/m | 10.81 eV (Reflects visible + near-UV) |
| Silver (Ag, 300 K) | n = 5.86e28 m^-3, tau = 40 fs | 66.05 MS/m (Highest metal!) | 8.99 eV (Free-electron edge) |
| Invalid Density or Tau <= 0 | Domain Guard | Guarded Input Error | "Invalid Drude Inputs" |
Drude DC Electrical Conductivity sigma in MegaSiemens per Meter (MS/m = 10^6 S/m)
Evaluates ((n_m3 * (1.6021766e-19^2) * (tau_fs * 1e-15)) / 9.1093837e-31) / 10^6.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Drude Inputs", ROUND(((B2 * (1.6021766e-19 ^ 2) * (B3 * 1e-15)) / 9.1093837e-31) / 1000000, 2))Free-Electron Bulk Plasmon Photon Energy E_p = hbar * omega_p (eV)
Multiplies reduced Planck constant hbar = 6.582119569e-16 eV*s by SQRT((n * e^2) / (eps0 * m_e)).
=IF(B2 <= 0, "Invalid Drude Inputs", ROUND(6.582119569e-16 * SQRT((B2 * (1.6021766e-19 ^ 2)) / (8.8541878e-12 * 9.1093837e-31)), 2))Try the example
Use the original values or change them to test your understanding.