Showing papers on "Positronium published in 2022"

Many-body theory of positron binding to polyatomic molecules

Abstract: Positrons bind to molecules leading to vibrational excitation and spectacularly enhanced annihilation. Whilst positron binding energies have been measured via resonant annihilation spectra for $\sim$90 molecules in the past two decades, an accurate \emph{ab initio} theoretical description has remained elusive. Of the molecules studied experimentally, calculations exist for only 6, and for these, standard quantum chemistry approaches have proved severely deficient, agreeing with experiment to at best 25% accuracy for polar molecules, and failing to predict binding in nonpolar molecules. The mechanisms of binding are not understood. Here, we develop a many-body theory of positron-molecule interactions and uncover the role of strong many-body correlations including polarization of the electron cloud, screening of the positron-electron Coulomb interaction by molecular electrons, and crucially, the unique non-perturbative process of virtual-positronium formation (where a molecular electron temporarily tunnels to the positron): they dramatically enhance binding in polar molecules and enable binding in nonpolars. We also elucidate the role of individual molecular orbitals, highlighting the importance of electronic $\pi$ bonds. Overall, we calculate binding energies in agreement with experiment (to within 1% in cases), and we predict binding in formamide and nucleobases. As well as supporting resonant annihilation experiments and positron-based molecular spectroscopy, the approach can be extended to positron scattering and annihilation $\gamma$ spectra in molecules and condensed matter, to provide fundamental insight and predictive capability required to properly interpret materials science diagnostics, develop antimatter-based technologies (including positron traps, beams and positron emission tomography), and understand positrons in the galaxy.

Perspectives on translation of positronium imaging into clinics

Abstract: The image of positronium properties created in the patient’s body during PET examination tells about the inter- and intra-molecular structure of the tissue and the concentration of bio-active molecules in the tissue [2–4]. In this article, we advocate the opinion that total-body PET systems, thanks to their high imaging sensitivity and high time resolution, open up the prospect of translating positronium imaging into clinics.

Discovery prospects for long-lived multiply charged particles at the LHC

Abstract: In this work, we aim to provide a comprehensive and largely model independent investigation on prospects to detect long-lived multiply charged particles at the LHC. We consider particles with spin 0 and $\frac{1}{2}$, with electric charges in range $1 \le |Q/e| \le 8$, which are singlet or triplet under $SU(3)_C$. Such particles might be produced as particle-antiparticle pairs and propagate through detectors, or form a positronium(quarkonium)-like bound state. We consider both possibilities and estimate lower mass bounds on new particles, that can be provided by ATLAS, CMS and MoEDAL experiments at the end of Run 3 and HL-LHC data taking periods. We find out that the sensitivities of ATLAS and CMS are generally stronger than those of MoEDAL at Run 3, while they may be competitive at HL-LHC for $3 \lesssim |Q/e| \lesssim 7$ for all types of long-lived particles we consider.

Positronium Lifetime Image Reconstruction for TOF PET

Abstract: Positron emission tomography is widely used in clinical and preclinical applications. Positronium lifetime carries information about the tissue microenvironment where positrons are emitted, but such information has not been captured because of two technical challenges. One challenge is the low sensitivity in detecting triple coincidence events. This problem has been mitigated by the recent developments of PET scanners with long (1-2 m) axial field of view. The other challenge is the low spatial resolution of the positronium lifetime images formed by existing methods that is determined by the time-of-flight (TOF) resolution (200-500 ps) of existing PET scanners. This paper solves the second challenge by developing a new image reconstruction method to generate high-resolution positronium lifetime images using existing TOF PET. Simulation studies demonstrate that the proposed method can reconstruct positronium lifetime images at much better spatial resolution than the limit set by the TOF resolution of the PET scanner. The proposed method opens up the possibility of performing positronium lifetime imaging using existing TOF PET scanners. The lifetime information can be used to understand the tissue microenvironment in vivo which could facilitate the study of disease mechanism and selection of proper treatments.

In-situ positron annihilation lifetime measurements of strained isoprene rubber filled with carbon black

Abstract: The free volume in carbon-black filled isoprene rubber is determined by in-situ positron annihilation lifetime measurements on the rubber subject to tensile strain using ortho-positronium as a probe. The dependence of free volume on the elongation and carbon black loading in the strained state is investigated to clarify the reinforcement mechanism of carbon black in synthetic rubber. Although no change in the ortho-positronium lifetime is detected, the ortho-positronium formation probability sharply decreases with increasing elongation and carbon black concentration in the strained state. The percent reduction is enhanced at high elongations and carbon black loadings. This result indicates the formation of both a bound rubber layer around the CB fillers and regions in the strained rubber phase with oriented molecular chains characterized by lower mobility, where ortho-positronium formation is suppressed.

Pseudovector and pseudoscalar spin-dependent interactions in atoms

Abstract: Hitherto unknown elementary particles can be searched for with atomic spectroscopy. We conduct such a search using a potential that results from the longitudinal polarization of a pseudovector particle. We show that such a potential, inversely proportional to the boson's mass squared, $V \propto 1/M^2$, can stay finite at $M \to 0$ if the theory is renormalizable. We also look for a pseudoscalar boson, which induces a contact spin-dependent potential that does not contribute to new forces searched for in experiments with macroscopic objects, but may be seen in atomic spectroscopy. We extract limits on the interaction constants of these potentials from the experimental spectra of antiprotonic helium, muonium, positronium, helium, and hydrogen.

Three-loop corrections to the Lamb shift in muonium and positronium

Abstract: We calculate hard spin-independent three-loop radiative corrections to energy levels in muonium and positronium which are due to radiative corrections with polarization insertions in two-photon exchange diagrams. These corrections could be relevant for the new generation of precise $1S\text{\ensuremath{-}}2S$ and $2S\text{\ensuremath{-}}2P$ measurements in muonium and positronium.

The Art of Positronics in Contemporary Nanomaterials Science: A Case Study of Sub-Nanometer Scaled Glassy Arsenoselenides

Abstract: The possibilities surrounding positronics, a versatile noninvasive tool employing annihilating positrons to probe atomic-deficient sub-nanometric imperfections in a condensed matter, are analyzed in application to glassy arsenoselenides g-AsxSe100−x (0 < x < 65), subjected to dry and wet (in 0.5% PVP water solution) nanomilling. A preliminary analysis was performed within a modified two-state simple trapping model (STM), assuming slight contributions from bound positron–electron (Ps, positronium) states. Positron trapping in g-AsxSe100−x/PVP nanocomposites was modified by an enriched population of Ps-decay sites in PVP. This was proven within a three-state STM, assuming two additive inputs in an overall trapping arising from distinct positron and Ps-related states. Formalism of x3-x2-CDA (coupling decomposition algorithm), describing the conversion of Ps-decay sites into positron traps, was applied to identify volumetric nanostructurization in wet-milled g-As-Se, with respect to dry-milled ones. Under wet nanomilling, the Ps-decay sites stabilized in inter-particle triple junctions filled with PVP replaced positron traps in dry-milled substances, the latter corresponding to multi-atomic vacancies in mostly negative environments of Se atoms. With increased Se content, these traps were agglomerated due to an abundant amount of Se-Se bonds. Three-component lifetime spectra with nanostructurally- and compositionally-tuned Ps-decay inputs and average lifetimes serve as a basis to correctly understand the specific “rainbow” effects observed in the row from pelletized PVP to wet-milled, dry-milled, and unmilled samples.

Using inverse Laplace transform in positronium lifetime imaging

Abstract: Positronium (Ps) lifetime imaging is gaining attention to bring out additional biomedical information from positron emission tomography (PET). The lifetime of Ps in vivo can change depending on the physical and chemical environments related to some diseases. Due to the limited sensitivity, Ps lifetime imaging may require merging some voxels for statistical accuracy. This paper presents a method for separating the lifetime components in the voxel to avoid information loss due to averaging. The mathematics for this separation is the inverse Laplace transform (ILT), and the authors examined an iterative numerical ILT algorithm using Tikhonov regularization, namely CONTIN, to discriminate a small lifetime difference due to oxygen saturation. The separability makes it possible to merge voxels without missing critical information on whether they contain abnormally long or short lifetime components. The authors conclude that ILT can compensate for the weaknesses of Ps lifetime imaging and extract the maximum amount of information.

Positron scattering from hydrogen atom in quantum plasmas: S-wave resonance states

Abstract: The effects of quantum plasmas (QP) on the S-wave resonance states of the positron-hydrogen system have been investigated by using the stabilization method. An effective potential, in the form of exponential cosine screened Coulomb potential, is used to describe the interactions among the charged particles in QPs. A number of S-wave resonance states associated with different thresholds of s-states of hydrogen atom [H(ns)] and s-states of positronium atom [Ps(ns)] are reported. For plasma-free cases, our results agree well with the results of other reliable calculations. Moreover, an in-depth study has been made to investigate the changes produced in the energy Er and width Γ of those states due to varying screening strength of QP. Our present results of energy and width of the positron-hydrogen system under QP associated with the higher-lying thresholds of hydrogen (H) and positronium (Ps) are reported for the first time in the literature.

Forward emission of positronium from nanochanneled silicon membranes

Abstract: Positronium beam formation and manipulation are required in several fundamental experiments. Efficient positron/positronium conversion in transmission configuration would offer important geometrical advantages over the reflection one for these applications. A novel type of transmission positron/positronium converters, which consists of silicon membranes with pass-through nanochannels, was produced and tested. The amount of forward emitted positronium was studied as a function of the thickness of the membranes and the nanochannel size. A maximum of, at least, $(16\ifmmode\pm\else\textpm\fi{}4)%$ of positrons implanted in $(3.5\ifmmode\pm\else\textpm\fi{}0.5)\ensuremath{-}\ensuremath{\mu}\mathrm{m}$-thick membrane with a nanochannel size of 5--8 nm were found to be forward emitted as positronium. A similar maximum amount of, at least, $(16\ifmmode\pm\else\textpm\fi{}5)%$, was found to be emitted from a membrane $(7.7\ifmmode\pm\else\textpm\fi{}1.3)\ensuremath{-}\ensuremath{\mu}\mathrm{m}$-thick with a nanochannel size of 7--10 nm. A preliminary evaluation shows that the maximum amount of forward emitted positronium with the entire kinetic energy distribution below 1 eV is, at least, 9% of the positrons implanted in the $(3.5\ifmmode\pm\else\textpm\fi{}0.5)\ensuremath{-}\ensuremath{\mu}\mathrm{m}$-thick membrane.

Comparative analysis of the Compton ionization of hydrogen and positronium

Abstract: The paper deals with the Compton disintegration of positronium and a comparison of the differential cross sections of this process with the similar cross sections in the case of the Compton ionization of the hydrogen atom. Special attention is paid to the resonances arising, when the electron and positron move parallel to each other in continuum states with the same velocities and zero relative momentum. It is likely that a manifestation of this effect in the double differential cross section (differential in the electron energy and angle) is found as an additional peak, which grows with a decrease in the photon energy.

Convolutional neural network-based reconstruction for positronium annihilation localization

Abstract: A novel hermetic detector composed of 200 bismuth germanium oxide crystal scintillators and 393 channel silicon photomultipliers has been developed for positronium (Ps) annihilation studies. This compact 4π detector is capable of simultaneously detecting γ-ray decay in all directions, enabling not only the study of visible and invisible exotic decay processes but also tumor localization in positron emission tomography for small animals. In this study, we investigate the use of a convolutional neural network (CNN) for the localization of Ps annihilation synonymous with tumor localization. Two-γ decay systems of the Ps annihilation from 22Na and 18F radioactive sources are simulated using a GEANT4 simulation. The simulated datasets are preprocessed by applying energy cutoffs. The spatial error in the XY plane from the CNN is compared to that from the classical weighted k-means algorithm centroiding, and the feasibility of CNN-based Ps annihilation reconstruction with tumor localization is discussed.

Conditions for obtaining positronium Bose–Einstein condensation in a micron-sized cavity

Abstract: Abstract The quest for making a triplet positronium (Ps) Bose–Einstein condensate confined in a micron-sized cavity in a material such as porous silica faces at least three interrelated problems: (1) About $$10^7$$ 10 7 spin polarized Ps atoms must be injected into a small cavity within a porous solid material without vaporizing it. (2) It is known that Ps atoms confined in 30–100 nm diameter cavities in porous silica do not remain in the gas phase, but become stuck to the cavity walls at room temperature (Cooper et al., Phys. Rev. B 97:205302, 2018). (3) Cooling a gas of Ps atoms to cryogenic temperatures by energy exchange with the walls would be a very slow process (Saito and Hyodo, in: Surko, Gianturco (eds) New Directions in Antimatter Chemistry and Physics, Springer Dordrecht, Netherlands, 2001) because of the relatively low collision rate with the walls and the large mismatch between the masses of the Ps and the wall atoms. A possible solution of these difficulties is presented, based on cooling the implanted positrons in an isotopically pure diamond single crystal target, subsequent saturating of the wall Ps coverage so that a substantial portion of the Ps will be in the gaseous state, and thermalizing the gas-phase Ps via collisions with the low effective mass wall Ps. A design process for the target material is outlined as well, including preliminary results in porous cavity fabrication using focused ion beam milling. Graphical abstract

Extended Positron–Positronium Trapping Defects in the MgAl2O4 Spinel Ceramics

Abstract: Extended free‐volume positron trapping defects (positron trapping channel) and nanopores (positronium decaying channel) in the initial and modified MgAl2O4 ceramics sintered at 1200, 1300, and 1400 °C are investigated using positron annihilation lifetime spectroscopy within four‐component fitting procedure. It is established that the number and size of extended defects near grain boundaries (described by the second component) decrease with increasing sintering temperature of ceramics, which correlates with the content of additional phases in the materials. The agglomeration of nanopores occurs in the initial ceramics (described by the third component) and the decrease of number and size of nanopores in the modified ceramics. The corresponding processes of the evolution of nanopores described by the fourth component are reflected mainly in their reduction, except for ceramics sintered at 1400 °C. Such changes are associated with the evolution of porous structure and increased contact area between the grains of ceramics.

Multi-photon time-of-flight MLEM application for the positronium imaging in J-PET

Abstract: We develop a positronium imaging method for the Jagiellonian PET (J-PET) scanners based on the time-of-flight maximum likelihood expectation maximisation (TOF MLEM). The system matrix elements are calculated on-the-fly for the coincidences comprising two annihilation and one de-excitation photons that originate from the ortho-positronium (o-Ps) decay. Using the Geant4 library, a Monte Carlo simulation was conducted for four cylindrical 22Na sources of β+ decay with diverse o-Ps mean lifetimes, placed symmetrically inside the two JPET prototypes. The estimated time differences between the annihilation and the positron emission were aggregated into histograms (one per voxel), updated by the weights of the activities reconstructed by TOF MLEM. The simulations were restricted to include only the o-Ps decays into back-to-back photons, allowing a linear fitting model to be employed for the estimation of the mean lifetime from each histogram built in the log scale. To suppress the noise, the exclusion of voxels with activity below 2% – 10% of the peak was studied. The estimated o-Ps mean lifetimes were consistent with the simulation and distributed quasi -uniformly at high MLEM iterations. The proposed positronium imaging technique can be further upgraded to include various correction factors, as well as be modified according to realistic o-Ps decay models.

Near-threshold behavior of positronium-antihydrogen scattering cross sections

Abstract: The authors numerically study the scattering of antihydrogen and positronium atoms, at collision energies just above the threshold for the rearrangement reaction resulting in antihydrogen positive ions. The formation of such ions is of interest for ongoing or planned experiments involving antimatter with antihydrogen at CERN.

Positron annihilation lifetime spectroscopy of adipose, hepatic, and muscle tissues

Abstract: Abstract Can positron annihilation lifetime spectroscopy (PALS) be employed to discriminate different types of soft tissues? This work reports our experimental measurements of the three components of positron annihilation lifetime spectra for three types of bovine, non-fixated soft tissues: adipose, hepatic, and muscle. These three components of PALS spectra correspond to contributions from annihilation lifetimes of para-positronium (p-Ps), positron, and ortho-positronium (o-Ps). We also report a benchmark comparison between PALS and X-ray phase-contrast imaging, which is the current state-of-the-art for soft tissue imaging and analysis. Our measurements showed that the joint probability of annihilation from p-Ps and positron in the soft tissues increases with electron density of the tissue and hence correlates well with the mean voxel values measured by X-ray phase-contrast computed tomography. Notably, the o-Ps lifetime in adipose tissue (2.53±0.01 ns) was approximately 25% longer than in hepatic (2.03±0.02 ns) and muscle tissues (2.00±0.01 ns). The significance here is that the o-Ps lifetime is a viable non-invasive probe for analyzing and discriminating the different soft tissues with a strong sensitivity to the lipid content of the tissue.

Bound-state properties, positron annihilation and hyperfine structure of the four-body positronium hydrides

Abstract: Bound-state properties of the ground (bound) -state(s) in the four-body positronium hydrides HPs, HPs (DPs), HPs (TPs) and MuPs are determined and investigated. By using numerical data from our computations of these four-body systems, we have determined a number of different annihilation rates for each of these positronium hydrides and evaluated the hyperfine structure splitting. The properties of the ground (bound) states in the four-body exitonic complexes, where and , have also been evaluated numerically. The neutral four-body systems with are similar to the HPs hydrides. In particular, each of these states has only one bound state. We also discuss applications of the exponential and semi-exponential variational expansions for accurate, bound state computations of the four-body positronium hydrides. GRAPHICAL ABSTRACT

Development of gamma-ray-induced positron age-momentum correlation measurement.

Abstract: In conventional positron annihilation spectroscopy using radioisotopes, source contributions are unavoidable since positrons annihilate in the material covering the radioisotopes. Part of the positrons annihilate within the radioisotopes even when radioisotopes are deposited directly on a sample. Gamma-ray-induced positron annihilation spectroscopy makes it possible to measure only the spectra of a sample without source contributions since positrons are directly generated inside the sample from the gamma rays by pair production and annihilate inside the same sample. In this study, a new positron age-momentum correlation measurement system using ultrashort pulsed gamma rays is developed. The gamma rays with an energy of 6.6 MeV are generated by the inverse Thomson scattering of laser photons by high-energy electrons and are irradiated to the sample. The laser pulse can fully control the timing of gamma-ray generation. This characteristic and the use of a digital oscilloscope with a 12-bit vertical resolution enable us to develop a simple measurement system. Time-resolved momentum distributions for stainless steel with no defects and deformed interstitial free steel show the explicit differences reflecting the type of defect; for BaF2 single crystals, the results have been interpreted by considering the formation of positronium.

Positron accumulation in the GBAR experiment

Abstract: We present a description of the GBAR positron (e+) trapping apparatus, which consists of a three stage Buffer Gas Trap (BGT) followed by a High Field Penning Trap (HFT), and discuss its performance. The overall goal of the GBAR experiment is to measure the acceleration of the neutral antihydrogen (H) atom in the terrestrial gravitational field by neutralising a positive antihydrogen ion (H+), which has been cooled to a low temperature, and observing the subsequent H annihilation following free fall. To produce one H+ ion, about 10^10 positrons, efficiently converted into positronium (Ps), together with about 10^7 antiprotons (p), are required. The positrons, produced from an electron linac-based system, are accumulated first in the BGT whereafter they are stacked in the ultra-high vacuum HFT, where we have been able to trap 1.4(2) x 10^9 positrons in 1100 seconds.

Energy dissipation of para-positronium in polymers and silica glass

Abstract: In this letter we discuss the energy dissipation of short-lived para-positronium (p-Ps) in polymers and silica glass. The S parameter characterizing the Doppler broadening of p-Ps annihilation is determined from the previously reported systematic data of positron annihilation age momentum correlation for various polymers and silica glass. A comparison of the S parameter with that expected for thermalized p-Ps trapped in a free volume reveals that p-Ps is not thermalized and possesses excess energy in fluorinated polymers and silica glass, indicating that it is difficult for Ps to lose energy in substances containing heavy elements such as fluorine and silicon.

Unraveling the Phase Behavior of Water Confined in Nanochannels through Positron Annihilation

Abstract: Water confined in cylindrical pores of silica with a diameter of 2 nm was studied by positron annihilation lifetime spectroscopy. Instead of freezing at 273 K, the glass-like transition through the viscous state to the solid phase was observed in the wide temperature range of 225–188 K. Reducing the vapor pressure over confined water allowed us to obtain the negative pressure pn = −164 MPa in liquid water, and the phase transition temperature range changed to 234–189 K. The temperature dependence of the ortho-positronium (o-Ps) lifetime in water under pn became consistent with the surface tension as in most liquids, as opposed to the anomalous temperature dependence in water under the saturated vapor pressure. This indicates a reduction in the number of hydrogen bonds in confined water under pn. On this basis, to explain the anomalous temperature dependence, we propose a hypothesis of a partial energy transfer to distant water molecules via the OH-stretching modes instead of repelling the molecules surrounding o-Ps. Our results indicate that thermally generated defects in the confined ice are accompanied by permanent defects, which predominate in the ice formed under pn. Both types of defects have the same volume of 70 Å3, which is much greater than in hexagonal ice Ih. We conjecture that the defects are larger due to the influence of the silanol groups on the structure of solid water. These observations imply that the structure of confined ice differs significantly from that of bulk ice and the vapor pressure influences its formation.

Enhancement of positron binding energy in molecules containing π bonds

Abstract: Observation of vibrational Feshbach resonances in the annihilation spectra of positrons on molecules provides a direct measurement of the positron-molecule binding energy ε B . Annihilation measurements are presented for ring hydrocarbons with different numbers of π bonds, for which it is observed that the presence of π bonds generally increases the positron binding energies. These molecules were chosen because other global molecular parameters (e.g., polarizability, dipole moment, and geometry) are approximately constant, so the observed differences in ε B can be related to changes in the nature of the bonds. The molecular ionization potential E i is an exception: for these molecules, the inclusion of π bonds tends to decrease E i , and the number of π bonds also exhibits a correlation with ε B . Comparison with other molecules with π bonds indicates that the changes in ε B are better correlated with the changing electronic structure of the bonds rather than with a direct dependence of ε B on E i . The relationship between the dependence of ε B on the number of π bonds and electron-positron correlation effects (such as virtual positronium formation) is discussed.

CPT symmetry test in positronium annihilations with the J-PET detector

Abstract: Discrete symmetry under combined transformation of charge, parity and time reversal (CPT) can be tested in the decays of positronium atom, the lightest bound system built of charged leptons. Jagiellonian Positron Emission Tomograph (J-PET) device constructed from plastic scintillators, detects the photons originating from electron positron annihilation. This feature enables J-PET to study CPT symmetry in the three photon annihilations of the triplet state of positronium. Signs of violation of the CPT symmetry can be sought as a non-vanishing expectation value of an angular correlation operator that is odd under CPT transformation. A technique to estimate the spin of ortho-positronium and momenta of annihilation photons for single recorded ortho-positronium annihilation events allows J-PET to measure the expectation value of a CPT symmetry odd angular correlation operator. J-PET measures a broad range of kinematical configurations of ortho-positronium annihilation to three photons and is the first experiment to determine the full range of the CPT-odd angular correlation.

Feasibility study of positronium application for blood clots structural characteristics

Abstract: Positron-electron annihilation in living organisms occurs in about 30% via the formation of a metastable ortho-positronium atom that annihilates into two 511 keV photons in tissues because of the pick-off and conversion processes. Positronium (Ps) annihilation lifetime and intensities can be used to determine the size and quantity of defects in a material’s microstructure, such as voids or pores in the range of nanometers. This is particularly true for blood clots. Here we present pilot investigations of positronium properties in fibrin clots. The studies are complemented by the use of SEM Edax and micro-computed tomography (µCT) to evaluate the extracted thrombotic material’s properties. µCT is a versatile characterization method offering in situ and in operando possibilities and is a qualitative diagnostic tool. With µCT the presence of pores, cracks, and structural errors can be verified, and hence the 3D inner structure of samples can be investigated.

Calculation of Low-Energy Positron-Atom Scattering with Square-Integrable Wavefunctions

Abstract: The variational method is applied to the low-energy positron scattering and annihilation problem. The ultimate aim of the investigation is to find a computationally economical way of accounting for strong electron–positron correlations, including the effect of virtual positronium formation. The method is applied to the study of elastic s-wave positron scattering from a hydrogen atom. A generalized eigenvalue problem is set up and solved to obtain s-wave positron–hydrogen scattering phase shifts within 8×10−3 rad of accepted values. This is achieved using a small number of terms in the variational wavefunction; in particular, only nine terms that depend on the electron–positron distance are included. The annihilation parameter Zeff is also calculated and is found to be in good agreement with benchmark calculations.

Detection of low-energy charged particles by channel electron multipliers

Abstract: Abstract Experimental determinations of the detection efficiency for positrons impacting a channel electron multiplier with incident energies between 0–1400 eV are presented. A log-normal dependence with energy is established and used to compute the positron-to-positronium detection efficiency ratio as a function of positronium energy, as required for determining quantities involving the ratio of positron and positronium rates. A log-normal energy-dependence is also observed in results of previous work with electrons, protons and ions.