Positronium formation models

Let us now consider further the use of the method of models in elastic positron-helium scattering, which is the sole open channel for positron energies below 17.8 eV, the threshold for ground state positronium formation. The total Hamiltonian of the system is... 

However, if electronic excitation is unimportant compared to positronium formation, aPs(E) aex(E), the maximum Ore model prediction is obtained from equation (4.36) as... 

Another model of positronium formation, the so-called spur model, was originally developed by Mogensen (1974) to describe positronium formation in liquids, but it has found some applications to dense gases. The basic premise of this model is that when the positron loses its last few hundred eV of kinetic energy, it creates a track, or so-called spur, in which it resides along with atoms and molecules (excited or otherwise), ions and electrons. The size of the spur is governed by the density and nature of the medium since these, loosely speaking, control the thermalization distances of the positron and the secondary electrons. It is clear that electrostatic attraction between the positron and electron(s) in the spur can result in positronium formation, which will be in competition with other processes such as ion-electron recombination, diffusion out of the spur and annihilation. 

Guha, S. and Mandal, P. (1980). Model potential approach for positron-atom collisions I. Positronium formation in ground state in alkali atoms Na, K, Rb and Cs using the distorted-wave approximation. J. Phys. B At. Mol. Phys. 13 1919-1935. 

Mogensen, O.E. (1974). Spur reaction model of positronium formation. J. 

Zhang, Z. and Ito, Y. (1990). A new model of positronium formation resonant positronium formation. J. Chem. Phys. 93 1021-1029. 

Positronium formation is a complex mechanism, not fully understood yet. Various models have been proposed. The first is the Ore model [Goldanskii, 1986] Ps formation occurs when a positron captures an electron (e.g., from a molecule M) according to the following scheme ... 

FIGURE 6. Spur reaction model for positronium formation. 

FIGURE 7, Modified spur reaction model. factors which control them, i.e. degree of order in the structure of dipolar molecules, intermolecular forces, polarization etc., will govern to a great extent the positronium formation. 

The formation of positronium atoms is affected by many factors but first of all, asO Eqs. (27.1) and O (27.2) suggest, by the material itself. According to the mentioned spur model (Mogenssen 1974), thermalized positrons compete with molecules of the material and with radiation products for available electrons. Sometimes, these spedes are such effective inhibitors or scavengers that they prevent positronium formation totally. Even if the inhibition is negligible, not all of the positrons can form Ps. Positronium formation reactions given by O Eqs. (27.2b) and (27.2c) require positrons of some particular energies. 

Independently of what one thinks of their formation mechanism, Ps atoms need space to be formed. In gasses, there is an enormous empty space between gas molecules, so space (or the shortage of it) is not a limiting factor of Ps formation. In fluids, however, Ps creates a small empty space, a bubble around itself. The bubble model was developed a long time ago (Ferrel 1957) and has been modified continually. Its present form tries to synthesize the results of the spur model and modern physical chemistry (Stepanov et al. 2000). In solids, structural free volumes might serve the empty space needed for positronium formation. 

The foundations of the Wannier treatment for positron impact are not as solid as for electron impact. The reason is that in the latter case, there are only two open channels near threshold elastic and inelastic scattering and ionization, whereas for positron impact, there is an additional, very strong channel, positronium formation. Furthermore, in the e impact case, the two light particles are much closer to one another in the final state, this casting doubt on the non-overlapping, classical orbits of the model. 

We have discussed mostly the temperature dependence of the positron mean lifetime tm. The temperature dependence of each separate lifetime component is more complex to review because the extraction of different lifetime components present in a spectrum is a delicate process. Using the two-state trapping model, most of the authors have limited the analysis of positron lifetime spectra to two lifetime components, after subtracting for source components and a low-intensity long component related to positronium formation at the surfaces. The first component, rj, is ascribed usually to annihilation of free positrons and the second, t2, to positrons trapped by defects. It should be pointed out that the relations frequently used to interpret the lifetime are based on the hypothesis that there is no positron detrapping, which is questionable at temperatures above 200 K. 

Positrons emitted for a radioactive source (such as 22Na) into a polymeric matrix become thermalized and may annihilate with electrons or form positronium (Ps) (a bound state of an electron and positron). The detailed mechanism and models for the formation of positronium in molecular media can be found in Chapters 4 and 5 of this book. The para-positronium (p-Ps), where the positron and electron have opposite spin, decays quickly via self-annihilation. The long-lived ortho positronium (o-Ps), where the positron and electron have parallel spin, undergo so called pick-off annihilation during collisions with molecules. The o-Ps formed in the matrix is localized in the free volume holes within the polymer. Evidence for the localization of o-Ps in the free volume holes has been found from temperature, pressure, and crystallinity-dependent properties [12-14]. In a vacuum o-Ps has a lifetime of 142.1 ns. In the polymer matrix this lifetime is reduced to between 2 - 4 ns by the so-called pick-off annihilation with electrons from the surrounding molecule. The observed lifetime of the o-Ps (zj) depends on the reciprocal of the integral of the positron (p+(rj) and electron (p.(r)) densities at the region where the annihilation takes place ... 

The mechanism of Mu formation is a matter of substantial importance. The scheme illustrated in Fig. 2 assumes that Mu is formed at the end of the charge-exchange cycles, and forms the basis of the epithermal model of Mu formation. However there is another view, a spur reaction model of Mu formation, in which it is assumed that i stops within or near the terminal spur of the muon track where it can combine with an e to form Mu. This model comes from an analogy with the same model for positronium (Ps) formation. Although the spur reaction model of Ps formation has received substantial support for condensed phases [51], the validity of the same model for Mu formation is still open to debate [521. 

Here Mu is assumed to be formed as a result of combination of and an excess electron. This view is the same as for the spur model of positronium (Ps) formation. While the spur model has received strong support for positronium yield in condensed phases, the validity of the same model for Mu formation is not clear. Figure 11 presents the original form of the spur model of Mu formation, since it helps to contrast the difference between the epithermal model (Fig. 2) and the spur model of Mu formation. Alternatively, the part of Mu formation, i.e., p and excess electron combination, in Fig. 11 may be replaced with the picture of Mu... 

Dauwe, C., Bas, C., and Palacio, C. A., Formation of positronium Multi-exponentials versus blob model. Radial. Phys. Chem., 76 (2), 280-284 (2007). 

A model which combines certain features of both models is Tao s "modified spur model" (37). In this model Tao considers both the possibility of combination of a positron with an electron created in the spur as well as the "direct" formation of a positronium, similar to the mechanism discussed in the Ore model, if the total kinetic energy of the resulting electron-positron pair is less than the potential energy between them. 

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