Positron Cooling and Annihilation in Noble Gases

Positron cooling and annihilation in room temperature noble gases is simulated using accurate scattering and annihilation cross sections calculated with many-body theory, enabling the first simultaneous probing of the energy dependence of the scattering and annihilation cross sections. A strikingly small fraction of positrons is shown to survive to thermalization: $\sim 0.1$ in He, $\sim 0$ in Ne, $\sim 0.15$ in Ar, $\sim 0.05$ in Kr, and $\sim 0.01$ in Xe. For Xe, the time-varying annihilation rate ${\overline{Z}}_{\text{eff}}(\tau )$ is shown to be highly sensitive to the depletion of the momentum distribution due to annihilation, conclusively explaining the long-standing discrepancy between gas-cell and trap-based measurements. Overall, the use of the accurate atomic data gives ${\overline{Z}}_{\text{eff}}(\tau )$ in close agreement with experiment for all noble gases except Ne, the experiment for which is proffered to have suffered from incomplete knowledge of the fraction of positrons surviving to thermalization and/or the presence of impurities.

Figure 1

(a) Momentum-diffusion coefficient $B(k)/k_{B}T=k{\sigma }_t(k)m/M$ vs positron momentum $k$. (b) $Z_{\text{eff}}(k)$ calculated using MBT including $s$-, $p$-, and $d$-wave positrons annihilating on valence and inner valence subshells of the noble gases. Vertical dashed line: thermal positron momentum $k_{\text{th}}=\sqrt{3k_{B}T}\sim 0.0528$ a.u. at $T=293\mathrm{K}$.

Figure 2

(a) Density plot of positron momentum distribution $f(k,\tau )$ for Ar, normalized as $\int f(k,\tau )dk=F(\tau )$, the fraction of positrons surviving (dashed-dotted line), calculated using 50 000 positrons initially distributed uniformly in energy up to the Ps-formation threshold. Also shown is the rms momentum (black dashed line) and thermal momentum $k_{\text{th}}=\sqrt{3k_{B}T}\sim 0.0528$ a.u. at $T=293\mathrm{K}$ (solid line). (b) Fraction of positrons surviving at time-density $\tau$ (solid lines). Also marked are the values of $F$ at the shoulder time ${\tau }_s$ (crosses) and the complete thermalization time ${\tau }_{\text{th}}$ (circles).

Figure 3

${\overline{Z}}_{\text{eff}}(\tau )$ for He to Xe calculated using $f(k,\tau )$ excluding and including loss of particles due to annihilation, initially distributed uniformly in energy (black dashed and solid lines, respectively), and including annihilation with initial energy equal to the Ps-formation threshold (magenta dotted line). Also shown for He: experiment of Coleman et al. [4] (red circles), present calculation with ${\overline{Z}}_{\text{eff}}$ scaled to the measured value of ${\overline{Z}}_{\text{eff}}=3.94$ (blue solid line), FP calculation of Campeanu [14] (blue dashed-dotted line), and model calculations of Boyle et al. [16] scaled to ${\overline{Z}}_{\text{eff}}=3.94$ (green dash-dash-dotted line); Ar: experiments of Coleman et al. [4, 8] (red circles) and FP calculation of Campeanu [8] (blue dashed-dotted line); Kr and Xe: experiment of Wright et al. [7] (red circles) and FP calculations of Campeanu [9] (blue dashed-dotted lines); Ne: experiment of Coleman et al. [4] (red circles) and FP calculation of Campeanu [8] (blue dashed-dotted lines). The calculated lifetime spectrum [i.e., observed annihilation rate $A(\tau )=-dF(\tau )/d\tau$] for Ne (blue staircase) is also shown (in arbitrary units), compared with experiment [4] (red circles). Black and red arrows mark the calculated and experimental shoulder lengths ${\tau }_s$.