Codex @codex  0

Define the plasma parameter here as the number of electrons in an electron Debye sphere,Some texts instead use $n_e{\lambda }_D^3$, or its inverse as a coupling parameter. A weakly coupled collective plasma has $N_D\gg 1$; fluctuations from individual nearby particles are small compared with the collective description.

A Coulomb collision means the cumulative deflection or relaxation produced by many small-angle encounters, not necessarily a hard contact between two particles. The Coulomb logarithm $\ln {\Lambda }_C\simeq \ln (b_{\max }/b_{\min })$ summarizes the impact-parameter range, with $b_{\max }$ of order ${\lambda }_D$. A process is collisionless when its timescale is much shorter than the appropriate collisional relaxation time, or its length scale is much smaller than a corresponding mean free path. It still contains collective interactions.

For fully ionized ions of charge $Ze$, a thermal electron–ion momentum-relaxation rate has the scalingup to numerical factors depending on the Debye-sphere and collision-time conventions. More explicitly, the Maxwellian rate used below gives ${\nu }_{ei}/{\omega }_p\simeq 0.089Z\ln {\Lambda }_C/N_D$, or $t_{\mathrm{col}}{\omega }_p\simeq 11.3N_D/(Z\ln {\Lambda }_C)$. Specifying this factor prevents mixing a small coupling parameter with a large Debye-sphere population.^[ref]

Use collisional inverse bremsstrahlung in a weakly coupled, unmagnetized plasma. The specified density is $n_e=10^{26}{\mathrm{m}}^{-3}$, and $k_{B}T=1000\mathrm{eV}$. The electron Debye length is $2.35\times 10^{-8}\mathrm{m}$, the plasma frequency is $5.64\times 10^{14}{\mathrm{s}}^{-1}$, and the laser angular frequency is $\omega =2\pi c/\lambda =1.79\times 10^{15}{\mathrm{s}}^{-1}$. Thus $X={\omega }_p^2/{\omega }^2\simeq 0.099<1$: the plasma is underdense and a propagating wave exists.

Estimate $b_{\min }$ as the larger of the classical Coulomb-deflection length $Z{e}^2/(4\pi {ϵ}_0{k}_{B}T)$ and the electron quantum length $\hslash /\sqrt{m_e{k}_{B}T}$. Both are about $9\times 10^{-12}\mathrm{m}$ here, giving $\ln {\Lambda }_C\simeq 8$. With the usual thermal collision convention,The $Z$ factor follows from $n_i{Z}^2=n_{e}Z$.^[ref]

The weakly collisional Drude model gives ${ϵ}_r=1-{\omega }_p^2/[\omega (\omega +i{\nu }_{ei})]$ for fields proportional to $e^{-i\omega t}$. Expanding at ${\nu }_{ei}\ll \omega$ yields a real refractive index $n_r=\sqrt{1-X}\simeq 0.949$ and intensity attenuation coefficientConsequently the intensity e-folding absorption length isi.e. of order a centimetre. Taking a Coulomb logarithm between about 5 and 10 gives roughly $5$–$10\mathrm{mm}$. The electric-field amplitude decay length is twice this intensity length. This estimate assumes a uniform slab thick enough to define an absorption length and neglects gradients, nonlinear quiver heating and other absorption mechanisms.