Absolute fraction of emitted Ps determined by geant4-supported analysis of gamma spectra

The fraction of positronium (Ps) emitted from a surface of a germanium single crystal at high temperature is usually assumed to approach unity at zero positron implantation energy. In the experiment, however, the determination of the absolute Ps fraction is not straightforward since recording a reference spectrum with $100\%$ Ps formation remains demanding. We use geant4-simulated detector responses to $2\gamma$ and $3\gamma$ radiation sources mimicking positron and Ps annihilation inside the (coincidence) Doppler-broadening spectrometer at NEPOMUC, FRM II, in order to derive a reliable value for the Ps fraction reemitted from a Ge(100) target heated close to its melting point. Analysis of the measured spectra by fitting the simulated spectra shows an absolute value of $72\pm 4\%$ maximum Ps formation, contradicting the $100\%$ assumption.

Figure 1

Exemplary gamma spectrum resulting from positrons implanted at $15\mathrm{keV}$ detected by a high-purity germanium detector. The important features of this spectrum are the main positron annihilation photo peak at $511\mathrm{keV}$, the pile-up peak at the right edge due to simultaneous detection of two such gamma quanta ($1022\mathrm{keV}$), the Compton edge at $340\mathrm{keV}$, and the valley region V, which is the most sensitive to $3\gamma$ annihilation of o-Ps.

Figure 2

Ps yield from pure Ge(100) heated to $1100\mathrm{K}$ as a function of positron implantation energy using the traditional $F$-parameter formalism as specified in the text. Measured values (symbols) were used for fitting the models describing the amount of implanted positrons diffusing back to the surface and forming Ps. The term $L_{+}$ is the positron diffusion length and $f_0$ is the maximum fraction of Ps formation following from thermal positron diffusion towards the surface. Note that $f_0$ can contain components of thermal and nonthermal Ps. Statistical error bars on the data are smaller than the symbols.

Figure 3

geant4-applicable sketch of the (coincidence) Doppler-broadening spectrometer at NEPOMUC as it was set up during the measurements on Ge(100) targets. In the center of the upper molybdenum disk, where the sample was located during the experiment, a point source of annihilation radiation is assumed, mimicking a pure $2\gamma$ spectrum and a continuous $3\gamma$ spectrum, respectively. Spectra as usually recorded by the surrounding upper detector were therefore simulated.

Figure 4

Simulated Ps annihilation spectra for the upper HPGe detector in the CDB spectrometer. Light blue (gray): Pure $2\gamma$ spectrum as from para-Ps or direct positron annihilations. Red (black): $3\gamma$ spectrum as from ortho-Ps annihilations. The dashed lines mark the energies $170\mathrm{keV}$ and $340\mathrm{keV}$, and hence $\frac{1}{3}$ and $\frac{2}{3}$ from the photo peak energy.

Figure 5

Measured spectra (light blue/gray) fitted with a superposition of simulated $2\gamma$ and $3\gamma$ spectra (red/black) for a positron implantation energy of $0.4\mathrm{keV}$ (top) and $28\mathrm{keV}$ (bottom). The positron-induced background $C_{BG}$ (dotted line) is the baseline on top of which the superimposed simulated spectra are derived. The amount of emitted Ps, $f_{\mathrm{Ps}}$, is derived from the ratio $N_{3\gamma }/N_{2\gamma }$, yielding 0.63 and 0.02, respectively.

Figure 6

Ps yields resulting from superposition of simulated $2\gamma$ and $3\gamma$ spectra, as demonstrated in Fig. 5, for all measured positron implantation energies (black squares). Fitting the data by a diffusion model of thermal positrons reproduces the expected positron diffusion lengths $L_{+}$ (solid black line). Applying a correction due to the limited flight path for o-Ps and consecutive pick-off annihilation, the absolute fractions of the emitted Ps are found (red circles and a thin line as a guide for the eye).