Thermodynamic state variables in quasiequilibrium ultracold neutral plasma

The pressure and internal energy of an ultracold plasma in a state of quasiequilibrium are evaluated using classical molecular dynamics simulations. Coulomb collapse is avoided by modeling electron-ion interactions using an attractive Coulomb potential with a repulsive core. We present a method to separate the contribution of classical bound states, which form due to recombination, from the contribution of free charges when evaluating these thermodynamic state variables. It is found that the contribution from free charges is independent of the choice of repulsive core length scale when it is sufficiently short-ranged. The partial pressure associated with the free charges is found to closely follow that of the one-component plasma model, reaching negative values at strong coupling, while the total system pressure remains positive. This pseudopotential model is also applied to Debye-Hückel theory to describe the weakly coupled regime.

Figure 1

Trajectories of an electron (blue lines) and ion (red circles) interacting through the Coulomb potential under conditions representing (a) free scattering and (b) a bound state. Arrows show the direction of electron motion from its starting point (blue dot). The ion-to-electron mass ratio is chosen to be ${10}^5$ for this case. Distances are in units of $a_N={10}^{-5}\mathrm{m}$.

Figure 2

Trajectories demonstrating various outcomes of three-body interactions between two electrons (red and black lines) and one ion (blue lines): (a) A classical electron-ion bound state interacts with an energetic electron, which frees the previously bound electron and forms a loosely bound state with the ion. (b) A classical electron-ion bound state interacts with an energetic electron, resulting in all free states. (c) An external electron interacts with a loosely bound state, gaining kinetic energy from the interaction and causing the bound pair to become more tightly bound. (d) An ion and two electrons all begin in a free state but form a bound pair via the three-body recombination. The ion-to-electron mass ratio is chosen to be 100 for this case. Distances are normalized by $a_N=1.3366\times {10}^{-5}$ m.

Figure 3

Radial distribution functions under weakly coupled conditions obtained from Eq. (4). (a) Electron-electron/ion-ion RDFs for $\mathrm{\Gamma }=0.02,0.5$. (b) Electron-ion RDFs for $\mathrm{\Gamma }=0.02$ and various $\alpha$. (c) Electron-ion RDFs for $\mathrm{\Gamma }=0.5$ and various $\alpha$ values. Like-species RDFs ($g_{ee/ii}$) have no $\alpha$ dependence, as they interact through the bare Coulomb potential.

Figure 4

Free-state (yellow region) and bound-state (cyan region) contributions to $g_{ei}\left(r\right)$ with $\mathrm{\Gamma }=0.5$ and $\alpha =0.1$. Horizontal and vertical lines represent $g_{ei}=1$ and $r_c$, respectively.

Figure 5

Fraction of bound states with respect to the repulsive core parameter $\alpha$ obtained from Eqs. (5) and (6). Lines use $g_{ie}$ from Debye-Hückel theory, Eq. (4), and filled blue circles use $g_{ie}$ from MD simulations for $\mathrm{\Gamma }=1$.

Figure 6

(a) Excess pressure under weakly coupled conditions calculated from Eqs. (1), (4), and (7). (b) Excess internal energy variation with $\alpha$ and $\mathrm{\Gamma }$ using Eq. (4). In both panels, dashed lines represent the combined free-plus-bound system, and solid lines represent just the free-charge contribution.

Figure 7

Dependence on $\mathrm{\Gamma }$ of (a) the excess pressure and (b) the excess internal energy at $\alpha =0.1$. Black lines with stars represent the total (bound-plus-free) system; blue lines with circles, free-charge contributions; and magenta line with squares, bound-state contributions only. Dashed red lines show the OCP values. Shaded regions indicate where the thermodynamic state variables are calculated using RDFs obtained from the Debye-Hückel description, while unshaded regions show the molecular dynamics results. Dotted lines represent an extension of Debye-Hückel theory into the strongly coupled regime. Blue-shaded regions show how the excess pressure and internal energy of the free charges vary when the right-hand side of Eq. (5) is varied from $exp\left(0.5\right)$ to $exp\left(2\right)$.

Figure 8

(a) Bound-state trajectories of electron-ion pairs during a simulation with $\mathrm{\Gamma }=1$ and $\alpha =0.1$ over a time interval of $3{\omega }_{pe}^{-1}$. Free particle trajectories have been removed. (b) RDFs under the same conditions, showing the peak in $g_{ei}\left(r\right)$ at $r=\alpha a$.

Figure 9

Electron temperature for free (dashed red line) and bound (blue line) species in a simulation with $\mathrm{\Gamma }=1$ and $\alpha =0.1$. The black line shows the total electron temperature in the system.

Figure 10

Evolution of the electron temperature in an ultracold plasma simulation at $\mathrm{\Gamma }=10$ and various values of $\alpha$. The heating rate increases as the repulsive core distance $\alpha a$ shrinks. The system is kept under the thermostat for the first 250 ${\omega }_{pe}^{-1}$ and then lifted to allow free temperature evolution.

Figure 11

Radial distribution functions $g_{ee}\left(r\right)$ (top) and $g_{ei}\left(r\right)$ (bottom) for equilibrium electron-ion plasma, showing variation with $\alpha$ (left) and $\mathrm{\Gamma }$ (right).

Figure 12

Dependence of (a) the excess pressure and (b) the excess internal energy on the repulsive core parameter $\alpha$ at various coupling strengths. Dashed lines indicate results for the combined free-plus-bound system and solid lines with symbols indicate results isolating the free-charge components of the system.