Experimental Measurement of Self-Diffusion in a Strongly Coupled Plasma

We present a study of the collisional relaxation of ion velocities in a strongly coupled, ultracold neutral plasma on short time scales compared to the inverse collision rate. The measured average velocity of a tagged population of ions is shown to be equivalent to the ion-velocity autocorrelation function. We thus gain access to fundamental aspects of the single-particle dynamics in strongly coupled plasmas and to the ion self-diffusion constant under conditions where experimental measurements have been lacking. Nonexponential decay towards equilibrium of the average velocity heralds non-Markovian dynamics that are not predicted by traditional descriptions of weakly coupled plasmas. This demonstrates the utility of ultracold neutral plasmas for studying the effects of strong coupling on collisional processes, which is of interest for dense laboratory and astrophysical plasmas.

Popular Summary

In strongly coupled plasmas, the Coulomb interaction energy between neighboring particles exceeds the thermal kinetic energy. Dense laboratory and astrophysical plasmas can be strongly coupled, such as in inertial confinement fusion experiments, white dwarf stars, and gas giant planet interiors. These strong interactions limit our ability to model and understand these systems because they lead to a breakdown of fundamental assumptions underlying the standard theoretical description of collision rates and transport coefficients. Here, we describe the first experimental study of short-time-scale, collisional dynamics in a bulk, strongly coupled plasma, including a new method to probe plasma dynamics and a measurement of self-diffusion.

We conduct our experiments with neutral plasmas that are among the coldest in existence, with temperatures barely 1 degree above absolute zero. These plasmas are created by ionizing an ensemble of laser-cooled ${}^{88}\mathrm{Sr}$ atoms. Strong coupling is accordingly obtained at relatively low density, which leads to slow dynamics and makes short-time-scale processes (compared with the inverse collision rate) experimentally accessible. An optical pump-probe technique that tracks the plasma ions isolates the effects of strong coupling on collisional processes. This physics has traditionally been studied using large-scale computer simulations, but direct comparisons of results with experiment have been lacking until now. We study the ion velocity distributions, and we show that our findings are consistent with molecular dynamics simulations.

The combination of atomic and plasma physics opens up a new direction in an area of plasma physics that has traditionally been the playground of astrophysics and large national facilities.

Figure 1

(a) Optical pumping and LIF spectroscopy. Ions are optically pumped from the $+1/2$ to $-1/2$ electronic spin state around $v_x=-{\mathrm{\Delta }}_p/k$, and from the $-1/2$ to $+1/2$ spin state around $v_x=+{\mathrm{\Delta }}_p/k$ with two counterpropagating, circularly polarized laser beams detuned by $-{\mathrm{\Delta }}_p\mathrm{rad}/\mathrm{s}$ from the ${}^2{S}_{1/2}-{}^2{P}_{1/2}$ transition of the strontium ion (421.7 nm). The velocity profiles of the individual spin populations are measured with LIF using a tunable circularly polarized probe beam of variable detuning ${\mathrm{\Delta }}_{\mathrm{pr}}$. (b) Idealized illustration of a pumped velocity distribution for $+1/2$ ions resulting from optical pumping (red curve), along with an unperturbed Gaussian thermal distribution (blue curve).

Figure 2

(a)–(c) Time evolution of ${}^2{S}_{1/2}-{}^2{P}_{1/2}$ LIF spectra, pumped (red circles) and unpumped (green squares), for spin $+1/2$ ions in a plasma with $\overline{\kappa }=0.55$ and ${\overline{\mathrm{\Gamma }}}_i=3.0$. (Bars denote time-averaged values; see Appendix B.) Spectra are fit (red and green lines) to a fifth-order Hermite-Gauss expansion of the velocity distribution convolved with the Lorentzian contribution from the laser and natural linewidths. (d)–(f) Full velocity distributions determined from the fits of pumped data to Eqs. (3) and (4), along with constituent Hermite terms $n=0–5$. Most of the velocity distribution is contained in the first few terms. Terms of order larger than zero decay in time as the distribution relaxes to a Maxwellian.

Figure 3

Relaxation of the average velocity of spin $+1/2$ ions in an optically pumped plasma. Lines indicate memory-function and exponential fits to early-time data, as well as a $t^{-3/2}$ fit to the late-time tail. (a) $⟨\mathrm{\Delta }{v}_x(t){⟩}_{+}/⟨\mathrm{\Delta }{v}_x(0){⟩}_{+}$ plotted versus time for two strongly coupled plasmas. (b) Data from (a) versus time scaled to the time integral of ${\omega }_p$. This plot shows the universal scaling of $⟨\mathrm{\Delta }{v}_x(t){⟩}_{+}/⟨\mathrm{\Delta }{v}_x(0){⟩}_{+}$ with ${\omega }_p$ for plasmas of different densities but with approximately the same ${\overline{\mathrm{\Gamma }}}_i$ and $\overline{\kappa }$. Panels (c) and (d) are the same plots as (a) and (b) but for a more weakly coupled plasma.

Figure 4

Early-time behavior of $⟨\mathrm{\Delta }{v}_x(t){⟩}_{+}/⟨\mathrm{\Delta }{v}_x(0){⟩}_{+}$. Lines indicate memory-function and exponential fits. Data and fits for plasmas with (a) $\overline{\kappa }=0.55$ and ${\overline{\mathrm{\Gamma }}}_i=3.0$ (${\overline{T}}_e=19K$), (b) $\overline{\kappa }=0.57$ and ${\overline{\mathrm{\Gamma }}}_i=3.2$ (${\overline{T}}_e=26K$), (c) $\overline{\kappa }=0.24$ and ${\overline{\mathrm{\Gamma }}}_i=2.1$ (${\overline{T}}_e=88K$), and (d) $\overline{\kappa }=0.59$ and ${\overline{\mathrm{\Gamma }}}_i=0.7$ (${\overline{T}}_e=15K$). Deviation from exponential decay is evident in more strongly coupled plasmas.

Figure 5

Plot of the normalized self-diffusion coefficient $D^{*}$ [Eq. (7))] calculated for our data. The solid blue and solid red lines represent results from MD simulations for $\kappa =0$ and $\kappa =0.6$, respectively [5, 6]. The black dashed line represents the Landau-Spitzer (LS) theory for weakly coupled plasmas, which diverges in the regime of strong coupling. The orange and the purple dashed lines use screened Coulomb and hypernetted chain interactions, respectively, as described in the text.

Figure 6

Typical evolution of (a) $T_i$, (b) $n_i$, (c) ${\omega }_p$, (d) ${\mathrm{\Gamma }}_i$, (e) $\kappa$, and (f) $T_e$ versus scaled time with smooth fits.

Figure 7

(a) Pumped and unpumped LIF spectra from regions of width $0.1{\sigma }_x$ across the plasma cloud spanning $x=\pm 0.2{\sigma }_x$ for the plasma described in Fig. 6. (b) Pumped and unpumped velocity distributions determined from the spectra shown in (a).

Figure 8

Evolution of average velocity of pumped spectra across all regions for the plasma described in Fig. 6. The black circles represent the mean value for each time point. The inset represents a closeup view of the same data over a scaled time ranging from 0 to 4.5 to highlight the nonexponential decay at early times.