Download S.V. Stepanov

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 +
=>
-
Physical Chemistry
=>
& Kinetics
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…)
…
Intratrack recombination
mechanism of Ps formation in
liquids instead of the Ore
model in gases:
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
e+qf + e-blob  e+…e- 
weakly bound (~0.1eV)
+e- pair.
stretched
e
 qf-Ps  Ps in a bubble
2. Because of energy loss on
+

weakly bound e e - pair
vibration, this pair transforms
rep  20 Å , Ebinding  0.1 eV
+
V
+V
e+


e
0
0
into a quasifree Ps, the
+e- in an
ground
state
of
e
e+ - e
E qf-Ps
binding
Coulombic attraction
~ 1-1.5 eV
unperturbed medium.
+
e e
quasi-free Ps
3. Further energy gain is due
rqf-Ps 3-6 Å
Ps in the
V V +V |+
bubble
to rearrangement of molecules
+Ry/2*(1-1/ )~
~ 3-5 eV (formation of the Ps bubble).
ee
ee+
EPs
What is a driving force of
-6.8 eV; Ps in vacuum
this process?

Ps
0
+
+
0
0
2
+ -
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 combined with the correct permutation symmetry of the e- wave function.
- eno HeH- ion
-2
0
2
-2
0
2
e

0
0
-5
-6
+
e-
2aB/r
-0.5
H atom
e-1
0
r/2aB
ss
e-
-2
ex
ce
e- e-
r/2aB
+
-4
energy /Ry
-3
He atom
xce
ss
-2
4aB/r
+e
energy /Ry
-1
bound
state
r/2aB
+
+
Without further details, Ferrel approximated this
repulsion by a potential well (rectangular). U
2
PR
(r)
U
EPs
R

(r)
EPs
R

RU
electronic
density
Ps bubble model => pick-off annihilation rate and
energetics and kinetics of the Ps bubble formation
The Tao-Eldrup Ps
bubble model relates
λpo=1/τ3 and R.
It assuming U=∞.
pick-off annihilation rate, 1/ns
Determination of δ in T-E Eq.
Relation between pick-off
annihilation rate of ortho-Ps and
surface tension.
Tao-Eldrup Eq. with fixed +=2 /ns
H O
gives =1.66 Å
0.6
2

D2O
CS2
OH
0.4 (CHC H) CO
4 9
3 2
C3H7OH
0.2
C6H6
C8H17OH
cC6H12
C2H5C6H5
1234TMB
dioxane
decane
C7H14
(CH3)2C6H4
C2H5OH
heptane
CH3CN
m-xylene
hexane
nC14H30 p-xylene
diethylether
C6F6 dodecane
toluene
n-C5H12 iC8H18
CH3OH
mesitylene
Xe
TMS
C(CH3)4
Ar
N2
H2
He
0
0
20
40
60
surface tension, dyn/cm
80
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
e2 => e2/ε, (ε≈n2≈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:
V0 is the
V0+ -? Same as for e-
ground state
energy of an
excess particle
in a medium
(work function)
How e+ and e- interacts each other in a medium
with a spherical cavity?
r=|r+-r-|
r
-
+

R  1
=
The energy of e+e- Coulombic
attraction may be expressed via
z
+
r+ series over the Legendre
polynomials Pl (cos θ):
2
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 = <Ψ+-|H|Ψ+-> → min over a and b
=> all energy contributions and
contact
density
and pick-off annihilation rate (λ+ ≈2/ns)
-Ry/ 2 =
=-1.7 eV

a
-Ry/ =
=-3.4 eV
2
delocalized e+e- pair
delocalized e+e- pair
b,Å
B
c and
1) Even if e+
e3
=
work functions
are
0
0 to 0, and
5 each 10
equal
0
free volume R, Å
particle does not
V++V-=2 eV; =2
consider a cavity
b a potential Ry/22=
neither as
aB as a poten. =1.7 eV
=nor
barrier,
1
2+V=
well, Ps bubble may -Ry/2
=0.3 eV
be formed due to ana
-Ry/2=
=-3.4 eV
enhancement of the
c=e+e(aB/a)
Coulombic
3
attraction
the
=inside
0
cavity
0 (no dielectric
5
10
0
free inside).
volume R, Å
screening
5
potential energy = 0
0
finite well, "point" Ps
total energy
energy, eV
=aB
Conclusions:
 =(a /a)
c ; a , Å ;
Ry/2
Ry/2 =
=1.7 eV
2
-5
Tao-Eldrup model
-Ry/2
-10
Coulombic energy
-Ry
5
10
free volume R, Å
e+e- kinetic energy
Ry/2
5
potential energy
finite well, point Ps
0
total energy
energy, eV
b
e+e- kinetic energy
delocalized e+e- pair
delocalized e+e- pair
b,Å
c ; a , Å ;
1
V++V-=0; =2
-5
Tao-Eldrup model
-Ry/2
-10
Coulombic energy
-Ry
5
10
free volume R, Å
Minimization of
e+e- energy, <H>
Two limiting cases:
1) “vacuum” Ps -- at
large R: a=aB ,
Etot=-Ry/2 , ηc=1 …
2) quasi-free Ps =
“vacuum” Ps with a
scaling e2 -> e2/ε, ε=2
a=εaB , ηc=1/ε3
Etot = -Ry/2ε2 +
+|V0++ V0- |
Big difference
with Tao-Eldrup!
λpick-off calculated according to Tao-
Eldrup, for a “point” Ps in a finite well
U and for e+e- pair
1
V +V+=0
V-+V+=2 eV
0.8
po (R) / +
2) At R<1.5-2.2 A all the obtained 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 localize, be a limiting factor of formation of the Ps bubble state.
U = 5.1 eV
U =0.96 Å
0.6
finite well
0.4 Tao-Eldrup
formula
0.2
qf-Ps
rep=3aB/21.5 Å
0
0
1
2
3
4
free volume radius R, Å
5
energy
0
-5
0
Tao-Eldrup
model

-Ry/2
total energy of the
Ps bubble
Ee+e-
delocalized e+e- pair
surface
3) For such a bubble
relative contact
V++Vthe
=0
density is ηc0.9. It is -higher then
experimental values surface
(0.65-0.75). This
energy
may indicate that e+ and e- interact with
0
a medium in a different way (i.e. V0total energy of the
+
>V0 ). Roughly,
Ps electron may be
Ps bubble
trapped in a cavity, but e+ will be bound
to e- by the Coulombic attraction. This
scenario can be considered
Ee+e-in the
frameworks of our approach. One may
-5
try psi-function of the pair in an
“asymmetric” towards e+ and e- form:
5
10 0
free volume R, Å

delocalized e+e- pair
V++V-=2 eV

energy, eV
Equilibrium size of Ps bubble in water is 5Å, which
is on 2 Å larger then in the Tao-Eldrup model
5
10
free volume R, Å
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, V0
-, of excess e- is a sum of its kinetic energy (exchange
repulsion) and polarization attraction [Springett B.E., Jortner J.,
Cohen M.N., 1968]
vacuum liquid phase
e -qf wave function
zero energy
in vacuum V

0
Up
If we consider qf-Ps in a continuum (no cavity), the
distance between e+ and e- is 3εaB/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 V0 - +V0 +,
U-out + U+out ≈-2 eV