(a) Derive the kinetic equation for the Langmuir occupation number.

(b) Using the approximations outlined in part (b) of Ex. 23.5, show that the Langmuir occupation number evolves in accord with the diffusion equation

where the diffusion coefficient is given by the following integral over the ionacoustic wave distribution:

(c) Discuss the strong similarity between the evolution law (23.59) for resonant Langmuir plasmons interacting with ion-acoustic waves, and the one for resonant electrons interacting with Langmuir waves [Eqs. (23.29c), (23.29d)].Why are they so similar?

Data from Exercises 23.5b.

(b) The ion-acoustic plasmons have far lower frequencies than the Langmuir plasmons, so ω_B ≪ ω_A ≈ ω_C. Assume that they also have far lower wave numbers, |k_B| ≪ |k_A| ≈ |k_C|. Assume further (as will typically be the case) that the ionacoustic plasmons, because of their tiny individual energies, have far larger occupation numbers than the Langmuir plasmons, so η_B ≫ η_A ∼ η_C. Using these approximations, show that the evolution law (23.57) for the ion-acoustic waves reduces to the form

where η_L is the Langmuir (waves A and C) occupation number, V_{g L} is the Langmuir group velocity, and R_C↔BA is the fundamental rate (23.56b).

Equation 23.56 b and 23.57.

Transcribed Image Text:

dni (k') dt = Vk [D(K) VKML (K)], . (23.59a)