Radial distribution function in Debye-Huckel theory

[CAF123]

Homework Statement

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.

Homework Equations

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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!

 

[mark726]

Thank you for your post. It seems like you are on the right track with your solution. To extract the explicit forms for $g_{++}(r)$ and $g_{-+}(r)$, we can use the definition of the radial distribution function you provided, $g(r) = e^{-\beta \phi(r) q}$. For $g_{++}(r)$, we can substitute in the Debye-Hueckel solution for the potential, which gives us $g_{++}(r) = e^{-\beta \frac{q}{4 \pi \epsilon} \frac{e^{-r/\lambda_d}}{r} q} = e^{-\frac{\beta q^2}{4 \pi \epsilon} \frac{e^{-r/\lambda_d}}{r}}$. Similarly, for $g_{-+}(r)$, we can substitute in the negative of the potential, which gives us $g_{-+}(r) = e^{-\beta \frac{q}{4 \pi \epsilon} \frac{e^{r/\lambda_d}}{r} q} = e^{\frac{\beta q^2}{4 \pi \epsilon} \frac{e^{-r/\lambda_d}}{r}}$.

Hope this helps! Let me know if you have any further questions.