Ore gap

Because of its relative simplicity, particular attention has been devoted to positronium formation in positron collisions with atomic hydrogen. Within the Ore gap the two open channels (other than direct annihilation) are... 

Among the most detailed and accurate investigations of positronium formation in the Ore gap are those of Humberston (1982, 1984) and Brown and Humberston (1984, 1985), who used an extension of the Kohn variational method described previously, see section 3.2, to two open channels. The single-channel Kohn functional, equation (3.37), is now replaced by the following stationary functional for the K-matrix ... 

Fig. 4.1. The results of various calculations of the l = 0 partial-wave contribution to the positronium formation cross section in positron-hydrogen scattering in the Ore gap A, Archer, Parker and Pack (1990) B, Humberston (1982) C, Stein and Sternlicht (1972) D, Chan and Fraser (1973) E, Wakid (1973) F, Dirks and Hahn (xlO) (1971) G, Wakid and LaBahn (1972) H, Khan and Ghosh (xlO-1) (1983) I, Born approximation (xl0 3).

The total positronium formation cross section in the Ore gap, constructed from the addition of accurate variational results for the first three partial waves and the values given by the Born approximation for all partial waves with l > 2, is plotted in Figure 4.4. On the scale of the ordinate, the s-wave contribution is too small to be visible. A very small s-wave contribution is found to be a feature of the positronium formation cross section for several other atoms. 

Fig. 4.4. The total positronium formation cross section in positron-hydrogen scattering in the Ore gap as calculated using the results of Brown and Humber-ston (1985) for / < 2 and the Born approximation for / >2.

Fig. 4.5. The angular distribution of positronium formation in positron-hydrogen scattering at various incident positron wavenumbers in the Ore gap.

Fig. 4.9. The total positronium formation cross section, and the various partial-wave contributions to it, for positron-helium scattering in the Ore gap. The contributions with l < 3 are determined variationally whilst the sum of all higher partial waves is calculated in the Born approximation.

Fig. 4.11. The angular distribution of positronium formation in positron-helium collisions at various energies in the Ore gap (Van Reeth and Humberston (1999b). The results for E = 17.8 eV, just above the positronium formation threshold, have been multiplied by a factor of 30.

Most other calculations of positronium formation in positron-helium scattering have employed much simpler methods of approximation, but results have usually been obtained over energy ranges extending well beyond the Ore gap. It must therefore be borne in mind that the experimental results include contributions from positronium formation into excited states as well as into the ground state. The Born approximation, used first by Massey and Moussa (1961) and subsequently by Mandal, Ghosh... 

This process is most effective when energy W of the positron lies within the interval named the Ore gap ... 

Coulombic attraction and increase average e+-e separation in the positronium. Both these factors decrease binding energy between the positron and electron constituting the positronium and therefore reduce the width of the Ore gap. 

Now let us estimate the low boundary, Wiow, of the Ore gap in molecular liquids. Because the Ore process is just an electron-transfer reaction, we assume that no rearrangement of molecules occurs and, therefore, the final positronium state will be quasi-free (formation of the bubble requires much longer time). The corresponding Born-Haber cycle is the following ... 

For example, in liquid water II = 8.8 eV, V w —1.2 eV. This relationship gives Wiow w 8-9 eV. It exceeds the lowest energy threshold Weoc 6.7 eV of electronic excitations, so in water, apparently, the Ore gap completely disappears. This takes place in the majority of molecular liquids, which explains the inefficiency of the Ore mechanism there. 

Over the last 30 years the recombination mechanism has become extremely widespread [16, 17]. It has been used to interpret extensive data on Ps chemistry, and explain variations of Ps yields from 0 to 0.7 in very different chemical substances where parameters of the Ore gap are practically the same. Variations of Ps formation probability under phase transitions have also received natural explanation. Experimentally observable monotonic inhibition of Ps yields (practically down to zero) in solutions of electron acceptors contradicts the Ore model, but is well incorporated in the recombination mechanism. It explains the anti-inhibition effect, including experiments on Ps formation in moderate electric fields in pure liquids and mixtures. 

Ps formation in polymers is not completely understood. Basically Ps can be formed in two ways. In the course of thermalization (energy loss) in polymers e passes through an energy region, called the Ore gap, where e can efficiently pick up an electron from the molecule M to fr>rm Ps as. 

Two basic models, the Ore gap model (20) and the spur reaction model (36), have been invoked to describe the Ps formation process. More recently several versions of a modified spur reaction model have been suggested (37-38). 

The lower boundary of this "Ore gap" is defined by the expression V - Ipg, where V is the ionization of the surrounding molecules and Ipg the ionization potential of Ps, 6.8eV. If the kinetic energy of the positrons exceeds V it is assumed... 

The Ore model is based on the simplified assumption that all positrons whose kinetic energy lies in the Ore gap produced Ps. In practice, however, the Ps formation process in the Ore gap has to compete with all other processes that can cause moderation of the positron to energies below the lower Ore Limit. The most important of these are elastic and inelastic collisions with substrate molecules, the energy transferred in the second case possibly stimulating molecular vibrations and rotations. This category also includes processes that lead to positron capture by addition of positrons to the substrate molecule AB. 

If the compound formation occurs above or within the Ore gap, the captured positrons are no longer available for the formation of Ps, whose yield is therefore decreased. 

The energy range most favorable for positronium formation is called Ore gap after its Norwegian discoverer (Ore 1949). The theory was developed for gasses, but, with some restrictions, it applies to liquids and solids as well. Ore has shown that there is a minimum energy, Emin> at which positronium formation is still possible ... 

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