Radiation-chemical aspects in the positron annihilation spectroscopy Serge Stepanov, Vsevolod Byakov (ITEP, Moscow) Gilles Duplâtre (see his review on the 2007 Polish seminar + many times before that): Conventional approach to description of Ps+S => Products reaction λ 3 exp (c S ) ≈ λ pick-off + kc S, k = 4π(D Ps +D S )(R Ps +R S ) = 2T/3η (2+R S /R Ps +R Ps /R S ), D S =T/6πηR S, D Ps =T/6πηR Ps These equations, with R Ps taken as the free Ps radius (0.53 A), delivers excellent radii of a solute S in a very large number of cases. Contrary, application of the theoretically well established concepts of the Ps bubble state (Tao-Eldrup; Buchikhin et al.) and time-dependent reaction rate coefficient k => k ij (t) = 4πD ij R ij [1+R ij /(πD ij t) 1/2 ] lead to unrealistic solute radii… G.Duplâtre: Many important unknowns still exist regarding Ps dynamics. The use of Ps in the bubble and time-dependent k remains questionable …

Answer: No. τ 3 must increase: T↑, σ ↓, R Ps ↑, τ pick-off ↑ More simple question: Can we explain an experimen- tally observed decrease of τ 3 vs. T in pure water in the frameworks of any conven- tional treatment (Tao-Eldrup or other Ps bubble model)? These annoying questions (simple, but fundamental) are really very important, because without getting clear answers on them the positron community will never be considered seriously by other scientific communities.

What to do ??? … Look deeper on radiation physics and chemistry of the intratrack processes and their mathematical description. For interpretation of the data on Ps interaction with dissolved scavengers (which G.Duplâtre refers) we should take into account Ps interaction with intratrack radiolytic species as well. This effect is not negligible. Because of e + * ionization slowing down (even in a chemically pure substance) many chemically active species are formed along/within the e + track. They interact with Ps (oxidation, stimulate spin conversion) and/or its precursor (track electron). In water following primary radiolytic products are formed: e - aq, H 2 O +, H +, OH, OH -, H 2, H 2 O 2. In the terminal part of the e + track (e + blob), where Ps is formed, lives and finally annihilates, there are about several tens of them.

Track structure of e + * and Ps formation

e+e+ e-e- Ps Intratrack mechanism of Ps formation: e + qf + e - blob  e + …e -   qf-Ps  Ps in a bubble Ps is formed in a bulk as a result of combination of the thermalized e + with one of the knocked out intrablob electrons. Initially Ps emerges in a medium as a weakly bound (~0.1eV) stretched e + … e - pair. Then this pair transforms into the quasi-free Ps, the ground state of nonlocalized e + e - pair in an unperturbed medium. Further energy gain is related with rearrangement of the molecules and formation of the Ps bubble state.

-- Mechanism of energy losses of e + and e -. Estimation of thermalization distances and times; -- Mechanisms of the Ps formation. Ps precursors. Quasifree-Ps state and its transformation to the Ps bubble state in liquids; -- Kinetics of the Ps bubble growth and energy dissipation accompanying Ps bubble formation; -- Effects of external and local electric fields; -- Role of trapped e- as Ps precursors at low T; -- Effect of local heating of the e + track caused by ionization slowing down of the energized e + and further recombination processes and so on… Let us leave aside (for a while) all the effects related (or foregoing) to the Ps formation: Now we shall focus on a kinetics of further intratrack chemical reactions of the Ps atom

In water different radiolytic species (OH, H 3 O + ) in the e + track may oxidize Ps. We will consider them in an independent pair approximation (a probability of a Ps+S i - reaction does not depend of the positions of the other S particles). Main primary radiolytic reactions in water:

(1) Track species have nonhomogeneous distribution. Do we already have an adequate formalism (even in the frame- works of the diffusion controlled reactions) for description of these processes? …ops, Not yet! - nonhomogeneous chemical kinetics approach (and the blob model) is based on the solution of the following equations: Ps reactions: Things should be taken into account where (bulk) rate constants k ij (t)=4πD ij R ij [1+R ij /(πD ij t) 1/2 ] are calculated separately, solving the Smoluchowski problem (as a flux on the absorbing sphere).

Fluxes on the absorbing sphere (and therefore the reaction rate constants) will be very different in this cases. Our case is Gaussian distribution of the species. So recalculation of k ij (t) is needed. This equation is rather empirical. It takes into account diffusion redistribution of reagents and their reactions when they meet each other at the same point of space. Usage of the Smolochowski rate constant is questionable because…

(2) We need a Ps bubble model to relate pick-off annihilation rate with the free volume radius of the Ps bubble. This radius will be recalculated to the hydrodynamic one, which determine Ps diffusion coefficient. => Again, it is important for interpretation of a particular experimental data, but it is a separate problem. We postpone it for a while. Ps reactions: Things should be taken into account So now we are approaching to revisiting a derivation of the kinetic equations and respective expressions for the rate constants from the very beginning. It seems to be a good tutorial task for this School.

…but before starting with this, a comment should be made concerning the difference of the diffusion coefficients of Ps atom and track species. In “conventional” approach it is used that D S;Ps =T/6πηR S;Ps, which is a based on 1) the Einstein relation between D and mobility μ: D=μT and 2) the Stokes formula for the friction force of a sphere moving in a viscous liquid (sticking boundary condition is assumed) μ S =6πηR S. But in case of Ps there is no sticking on the boundary (Rybchinski, 1911; Landau & Lifshiz, v.VI, p.99), it leads to μ=4πηR Ps and D Ps =T/4πηR Ps

Formally the problem reduces to calculation of the survival probability of one particle A (=Ps) surrounded by many reactive species B (=oxidizers) in the case when initial distribution of B’s is Gaussian. In 70-80’s M.Tachiya treated similar problem in a very general way. He derived equation for the pair survival probability (the adjoint Fokker-Planck equation) and solved it when many B’s uniformly distributed around one A (but not for the Gaussian distribution). But his method is difficult to follow, so we have followed to a simpler one suggested by A.Pontryagin et al. and A.Kolmogorov et al. (1933), which is still sufficient for our needs. To the theory of diffusion-controlled reactions: Formulation of the reaction rate constant in terms of the pair survival probability