Probing Positron Cooling in Noble Gases via Annihilation $\gamma$ Spectra

$\gamma$ spectra for positron annihilation in noble-gas atoms are calculated using many-body theory for positron momenta up to the positronium-formation threshold. These data are used, together with time-evolving positron-momentum distributions determined in the preceding Letter [Phys. Rev. Lett. 119, 203403 (2017)], to calculate the time-varying $\gamma$ spectra produced during positron cooling in noble gases. The $\gamma$ spectra and their $\overline{S}$ and $\overline{W}$ shape parameters are shown to be sensitive probes of the time evolution of the positron momentum distribution and thus provide a means of studying positron cooling that is complementary to positron lifetime spectroscopy.

Figure 1

$\gamma$ spectra $w_k(\epsilon )$ (normalised to unity at $\epsilon =0$) for annihilation in Ar, for positrons of momenta $k=0.04\mathrm{a}.\mathrm{u}.$ (solid line), $k=0.2\mathrm{a}.\mathrm{u}.$ (dashed line), $k=0.4\mathrm{a}.\mathrm{u}.$ (dashed-dotted line), and $k=0.6\mathrm{a}.\mathrm{u}.$ (dash-dash-dotted line).

Figure 2

$\gamma$ spectra $w_k(\epsilon )$ for positron annihilation in Ar and Xe as a function of positron momentum $k$ up to the positronium-formation threshold. Also shown are projections at Doppler-shifted energies $\epsilon =0$, 2, 4, 6, and 8 keV and at $k=0.02$ a.u. and from $k=0.1$ in step sizes of 0.1 a.u.

Figure 3

The shape parameter $W(k)$ (see the text) for raw (dashed line) and detector-convolved $\gamma$ spectra (solid line) for positron annihilation in He (blue), Ne (green), Ar (magenta), Kr (red), and Xe (black). Crosses mark the values for the thermally averaged (at $T=293\mathrm{K}$) detector-convolved spectra.

Figure 4

AMOC spectrum (number of $\gamma$ rays detected per unit time and Doppler-shifted energy) for Ar, in units of $\pi r_0^{2}c$, calculated using the detector-convolved spectra ${\overline{w}}_{\tau }(\epsilon )$, with positrons initially distributed uniformly in energy. Also shown are projections at $\tau =100$, 200, 300, 400, and 500 ns and at $\epsilon =0$, 1, 2, 3, 4, and 5 keV.

Figure 5

(a) $\overline{S}(\tau )$ for Ar: experiment [12] (red circles); present calculation, for positrons distributed uniformly in energy (thin solid line); that scaled as ${\overline{S}}_{\mathrm{sc}}=1.43\overline{S}-0.13$ (thick solid line); and calculated by first subtracting $0.1{\mathrm{keV}}^{-1}$ from ${\overline{w}}_{\tau }(\epsilon )$ (green dashed-dotted line). Also marked are the calculated [18] and measured [44] shoulder lengths (vertical dot-dashed and dashed lines, respectively) and calculated complete thermalization time (solid vertical line) [18]; (b)–(f) $\overline{W}(\tau )$ (red) and $\overline{S}(\tau )$ (blue) parameters for detector-convolved $\gamma$ spectra: excluding and including the depletion of the distribution due to annihilation, for positrons initially distributed uniformly in energy (dashed and solid lines) and with the energy equal to the Ps-formation threshold (dotted and dash-dotted lines, which are almost indistinguishable). Also shown are the shoulder lengths and thermalization times calculated in Ref. [18] (dashed and solid vertical lines) and the experimental shoulder lengths of Refs. [44] (He and Ne) and [20] (Ar, Kr and Xe) (dash-dotted vertical line). The steady-state values of $\overline{W}$ $(\overline{S})$ are He, 0.102 (0.610); Ne, 0.2067 (0.473); Ar, 0.624 (0.075); Kr, 0.664 (0.058); and Xe, 0.708 (0.043). The value for Ne neglected the depletion of the distribution due to annihilation. For Ne, at $\tau \sim 5000\mathrm{ns}\mathrm{amg}$, where $F(\tau )\lesssim {10}^{-2}$, $W(\tau )=0.2073$ and $S(\tau )=0.472$.