Debye Model

At sufficiently high temperature and low density, the hydrogen plasma behaves like a system of free electrons and protons, which interact via a screened Coulomb potential (Ebeling et al., 1976; Fowler and Guggenheim, 1965). The screening arises from a cloud of opposite charge of the size of the Debye radius $r_{\rm D}$. Assuming full dissociation, it is given by,

$$r_{\rm D}= \frac{1}{\kappa} \quad,\quad \kappa^2 = \frac{\beta}{\epsilon_0} \sum_{{\rm species}~i}{n_i Q_i^2}.$$ (269)

The screening leads to the following corrections $u_{D}$ and $p_{D}$ that are added internal energy and pressure of non-interacting Fermi gas,

$$u_{\rm D}= \frac{\kappa^3}{8 \pi \beta n} \quad,\quad p_{\rm D}= \frac{\kappa^3}{24 \pi \beta}\quad.$$ (270)

If Fermi statistics is not important the Debye corrections can be expressed in the terms of the coupling parameter $\Gamma$ (Eq. 1.4),

$$\frac{E_{\rm D}}{E_{\rm id}} = \frac{\Gamma^{3/2}}{\sqrt{6}}... ...rac{p_{\rm D}}{p_{\rm id}} = \sqrt{2/3} \; \Gamma^{3/2} \quad,$$ (271)

where $E_{\rm id}$ and $p_{\rm id}$ are the internal energy and pressure of a ideal gas of distinguishable particles. The Debye screening represents the first correction to the free particle behavior due to interactions in the limit of high temperature and low density. For small values of $\Gamma$, the Debye model is a reliable approximation. One finds deviations of less than $20\%$ in pressure and energy for $\Gamma < 0.5$ in discussed density range. However, at sufficiently high $\Gamma$, the Debye model overestimates the screening drastically and predicts a too small $r_{\rm D}$, which leads to unphysically low, even negative pressures.