Beyond the point Ps approximation (account of internal Coulombic attraction) S.V. Stepanov, D.S. Zwezhinski, V.M. Byakov Institute for Theoretical and Experimental Physics, Moscow -- Phylosophy of science: role of basic elementary models -- Intratrack mechanism of Ps formation -- Ps bubble: Exchange repulsion or something else? -- Non-point Ps. Account of intrinsic Coulombic e + e - interaction in a medium with a cavity. Ĥ via e + and e - work functions (+ polarization corrections) -- e + e - psi-function, minimization of total energy -- E ( R ), pick-off rate, contact density, equilibrium radius … -- How to link macroscopic and microscopic considerations?

What are general basis of a science? (take Quantum Mechanics as an example) 1) ideas/concepts (Ψ-function, operators, commutative properties, the Schrodinger equation, superposition principal...); 2) methods which realize ideas (perturbation theory, variational principle, quasiclassical approximation...) 3) basic elementary models illustrate how these methods do really work (oscillator, H-atom, potential well, free electron gas, the Thomas-Fermi model,...) 4) experimental verification (“Great experiments in physics”) Basic models are very important because: -- being simple and solvable they give us directions of thinking to conceive more complicate systems; -- they make up a basement of all education process

Quantum Mechanics Radiation + Positron + Ps Chemistry basic models are: - ionization stopping power; H.Bethe’s formula (1930) - ion-electron recombination; L.Onsager’s formula (1937) - diffusion-controlled rate constant; M.Smoluchowski (1917) - the Debye-Huckel screening (1923) - ambipolar diffusion; prescribed diffusion; G.Jaffe (1913) - energy losses of subexcitation electrons; H.Froehlich (1953) - model of solvated and quasifree electron; J.Jortner (1968…) - Ps bubble model; Ferrel, Goldanskii, Tao, Eldrup et al. (1957…) - … Physical Chemistry & Kinetics +=>

Intratrack recombination mechanism of Ps formation in liquids instead of the Ore model in gases: e + qf + e - blob  e + …e -   qf-Ps  Ps in a bubble 1. Ps is formed as a result of combination of the thermalized e + with one of the knocked out intratrack e - in the e + blob. Initially Ps appears as a weakly bound (~0.1eV) stretched e + e - pair. 2. Because of energy loss on vibration, this pair transforms into a quasifree Ps, the ground state of e + e - in an unperturbed medium. 3. Further energy gain is due to rearrangement of molecules (formation of the Ps bubble ). What is a driving force of this process?

In 1956(7) R.Ferrel suggested that the driving force of the formation of the Ps bubble is an exchange repulsion between the Ps electron and molecular electrons. Nature of the exchange repulsion is e - e - Coulombic repulsion com- bined with the correct permutation symmetry of the e - wave function. Without further details, Ferrel approximated this repulsion by a potential well (rectangular). U ( = depth) ~ 1-4 eV.

Ps bubble model => pick-off annihilation rate and energetics and kinetics of the Ps bubble formation The Tao-Eldrup Ps bubble model relates λ po =1/ τ 3 and R. It assuming U= ∞.

Relation between pick-off annihilation rate of ortho-Ps and surface tension. Determination of δ in T-E Eq.

However, beyond exchange repulsion there is important variation of internal Coulombic energy of e + e - pair (attraction between e+ and e- screened by the medium). Known scaling property of the Schrödinger Eq. for Ps atom e 2 => e 2 / ε, ( ε≈n 2 ≈2 ) gives for e + e - binding energy It is seen that in comparison with the exchange repulsion (typically U =1-4 eV) variation of the Coulombic energy is not small. So it should be taken into account. It was not done yet. We have to reject consideration of Ps as a point particle.

Hamiltonian of the e + e - pair in a medium with a spherical cavity Interaction of e + and e - with a medium: V 0 is the ground state energy of an excess particle in a medium (work function) V 0 + -? Same as for e -

How e + and e - interacts each other in a medium with a spherical cavity? The energy of e + e - Coulombic attraction may be expressed via series over the Legendre polynomials P l (cos θ):

Examples of e + and e - Coulombic interaction in a medium with a spherical cavity We really gain a lot of energy (several eV) only in the case when both particles are well inside the sphere (left figure).

Trial wave function for energy minimization (simplest, but sufficient) Total energy = → min over a and b => all energy contributions and contact density and pick-off annihilation rate (λ + ≈ 2/ns)

Two limiting cases: 1) “vacuum” Ps -- at large R : a = a B, E tot = - Ry/2, η c = 1 … 2) quasi-free Ps = “vacuum” Ps with a scaling e 2 -> e 2 / ε, ε =2 a = εa B, η c = 1/ε 3 E tot = - Ry/2ε | V V 0 - | Big difference with Tao-Eldrup! Minimization of e + e - energy, Conclusions: 1) Even if e+ and e- work functions are equal to 0, and each particle does not consider a cavity neither as a potential barrier, nor as a poten. well, Ps bubble may be formed due to an enhancement of the Coulombic e+e- attraction inside the cavity (no dielectric screening inside).

λ pick-off calculated according to Tao- Eldrup, for a “point” Ps in a finite well U and for e + e - pair 2) At R < A all the obtain- ed dependencies have a plateau, but at larger R there are significant variations. It is related with the known quantum mechanical phenomenon -- absence of a bound state of a particle in a small finite 3d-potential well. In such cavities Ps cannot be bound, it does not exert any repulsive pressure on them and does not stimulate their transformation into equilibrium Ps bubble. It could be that finding a suitable preexisting cavity, where qf-Ps may loca- lize, be a limiting factor of for- mation of the Ps bubble state.

Equilibrium size of Ps bubble in water is 5 Å, which is on 2 Å larger then in the Tao-Eldrup model 3) For such a bubble the relative contact density is η c  0.9. It is higher then experimental values ( ). This may indicate that e+ and e- interact with a medium in a different way (i.e. V 0 - > V 0 + ). Roughly, Ps electron may be trapped in a cavity, but e + will be bound to e - by the Coulombic attraction. This scenario can be considered in the frameworks of our approach. One may try psi-function of the pair in an “asymmetric” towards e + and e - form:

4) Challenge for the positron/Ps-newcomers : Any Ps bubble model reduces the original multi-particle problem to a simpler one: to a problem of one or two species in an external self-consistent field, describing Ps-medium interaction. To obtain this field we use macroscopic approaches. However, their linkage with, for example, actual arrangement of molecules, forming boundary of the Ps bubble, usage a concept of dielectric permittivity and so on, remains always uncertain. More adequate approaches should be developed or used...

In condensed phase in addition to the exchange repulsion there is polarization attraction between excess e - and a medium. So the ground state energy, V 0 -, of excess e - is a sum of its kinetic energy (exchange repulsion) and polarization attraction [ Springett B.E., Jortner J., Cohen M.N., 1968]

If we consider qf-Ps in a continuum (no cavity), the distance between e+ and e- is 3εa B /2 ≈1.5 Å (about the Wigner-Seitz radius of a molecule), so other molecules “see” this qf-Ps as an electrically neutral object. Therefore, it is reaso-nable to subtract from V V 0 +, U - out + U + out ≈-2 eV