# Radial distribution function in Debye-Huckel theory

1. Mar 1, 2015

### CAF123

1. The problem statement, all variables and given/known data
For a plasma containing two ionic species with opposite charges but the same density $n_{\infty}$, calculate the radial distribution functions $g_{++}(r), g_{-+}(r),$where $n_{\infty}g_{ij}(r)$ is the conditional probability density for finding a particle of type $i$ in a small volume at distance $r$ from one of type $j$. You may assume Debye-Hueckel theory is valid and should use the result that $$\lambda_D^2 = \frac{\epsilon kT}{2q^2 n_{\infty}}$$ where $\epsilon$ is a dielectric constant and $q$ is the charge of the species.

2. Relevant equations

radial distribution function $g(r) = e^{-\beta \phi(r) q}$

3. The attempt at a solution

Can write the number density of particles from some reference particle at the origin, $n(r) = n_{\infty} [ e^{q\beta \phi} + e^{-q\beta\phi}] = 2n_{\infty} \cosh (q \beta \phi)$. The solution to the Debye Huckel equation is that $$\phi = \frac{q}{4 \pi \epsilon} \frac{e^{-r/\lambda_d}}{r},$$ where $\lambda_D$ is given . I think it makes sense that the $g_{-+}(r)$ would be more sharply peaked or not as suppressed as $g_{++}(r)$ , but I am not quite sure how to extract the explicit forms.

Thanks!

2. Mar 6, 2015