Question

In the Lorentz model, take N 1028 m-3 for the density of bound electrons in an insulator, and a single transition at (00-6 × 1015 rad/sec (in the UV), and damping γ_a0/5 (quite broad). Assume Eo is 104 V/m. For three frequencies i) ω-ω0-2γ ii) ω-ω0, and iii) ω-ω0 + 2γ with the driving field E,ei(kr-at), find (a) the amplitude and phase of the susceptibility χ(w) and (b) n and k at the three frequencies.

Answer 1

N = 10^28 m^-3 w_0 = 6 times 10^5 rad/sec gamma = w_0/5 = 1.2 times 10^5 E_0 = 10^4 V/m (i) w = w_0 - 2 gamma = (6 - 2.4) times 10^5 = 3.6 times 10^5 (ii) w = w_0 = 6 times 10^5 (iii)w = w_0 + 2 gamma = 8.2 times 10^5 w_p = 10^28 times (1.6)^2 times 10^-28/8.854 times 10^-12 times 9.1 times 10^-31 = 0.0317 times 10$13 rad/sec Susceptibility (x) x(w = w_0 - 2 gamma) = w_p^2/w_0^2 - w^2 - jw gamma = (0.0317 times 10^43)^2/36 times 10^10 - 12.96 times 10^20 - j (3.6 times 10^5) times (1.2 times 10^5) = 1.00989 times 10^-3 times 10^86/10^10{(23.04) - j 4.32} = (1.00989 times 10^73)(23.09 + j 4.32)/(23.04 - j 9.32)(23.04 + j 4.32) = (23.15 + j 9.34) times 10^73/530.84 + 18.66 = (23.15 + j 4.39) times 10^73/599.50 \r\n x(w = w_0) = w_p^2/w_0^2 - w^2 - j w gamma = (0.0317 times 10^43)^2/36 times 10^10 - 36 times 10^10 - j (3.6 times 10^5) times 1.2 times 1.^5