Materials Science Forum Vols. 363-365 (2001) pp. 392-394
© 2001 Trans Tech Publications, Switzerland
Electric Field Effect on Ps Formation. Black Blob Model
Sergey V. Stepanov 1, Vsevolod M. Byakov 1
Cai-Lin Wang 2, Yoshinori Kobayashi 2, Kouichi Hirata 2
1
2
Institute of Theoretical and Experimental Physics, Moscow 117218, Russia
National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan
Keywords: electric field, Ps formation, blob model, trapping, intratrack processes, liquids
Abstract New version of the blob model for Ps formation is developed. It explicitly takes account of the
polarization interaction between the thermalized positron and its terminal blob.
Onsager’s model of geminate ion-electron recombination widely used in radiation chemistry is inadequate
for consideration of the electric field effect on Ps formation in molecular liquids [1], because the Ps atom is
formed in the terminal positron blob, containing many ( n0  20-30) ion-electron pairs, but not in an ordinary
spur. So in [2,3] we generalized the blob model of Ps formation by neglecting the polarization interaction of
the thermalized positron and its blob (“white blob” model). We adopted this assumption because of two opposite effects: 1) outdiffusion of intrablob electrons makes them to reside in an outer region of the blob,
which results in appearance of an excess positive charge in its central region, repelling e + from the blob; 2)
the presence of e+ within the blob may lead to rearrangement of intrablob electrons, which may decrease the
total energy of the system because of the Debye screening.
When e+ is outside the blob ( r  abl ; the blob size, abl , is about several tens of Å) their mutual interaction becomes less than temperature, T. However, by combining a simplified model [3] describing the spatial
distribution of intrablob electrons with an assumption that e+ resides in the center of the blob the total gain of
electrostatic energy is found to be about Tn01 / 3 rc abl  1 eV ( rc is the Onsager radius). Crude assessment of
the Debye screening energy, e 2 2r , (this expression is valid when the average distance r between intrablob species is less than the respective Debye radius) gives the same result.
Thus polarization interaction between the thermalized positron and its terminal blob may result in e + trapping within the blob. This effect may be responsible for the Ps yield reaching a plateau at fields D about
100 kV/cm, which was observed in polymers [4]. At these fields the white blob model [2,3] predicts lower Ps
yields than available experimental data. It should be noted that the region D  100 kV/cm lies within the
applicability of the theory, which ignores e+ heating by the field. In condensed molecular media the effect of
e+ heating becomes important at D  1 MV/cm.
In the present paper we describe another version of the blob model (“black blob” approach), which explicitly takes account of the above mentioned polarization interaction between the thermalized positron and
the blob. After the last ionization event the positron, having some excess energy, may easily escape from the
blob in contrast to the intrablob electrons which are held by ions. So e + should thermalize primarily outside
the blob. External electric field shifts aside e+ spatial distribution during its thermalization and later on, while
it resides in a quasi-free state. Below we distinguish between the positrons which finally become thermalized

inside ( e in ) and outside ( e out
) their blobs. The black blob model suggests that e in , which is confined within
the blob, forms Ps with one of the intrablob quasi-free electrons according to the reaction e in  e -  Ps .
Low fields (  100 kV/cm) can not affect the behavior of e in . To pull it out of the blob one needs higher
field (at least it should satisfy the following condition: eEabl  0.1 eV; below we restrict ourselves to the

case of low electric fields). Outside the blob e out
moves in a diffusion way and is exposed to the field, which


shifts e out
distribution aside and therefore reduces the Ps formation probability, PPs . When e out
reaches the
blob it becomes trapped and converts to e in .
The main assumption of the black blob model is that e+ (originally thermalized within the blob or entered
it later) can not escape from the blob in the form of the free positron. From mathematical viewpoint it implies that the prescribed diffusion method, which underlay our previous approach [2,3], can not be used for
Materials Science Forum Vols. 363-365
393

calculation of PPs now. So, as a first step, let us calculate the diffusion flux J of e out
into the blob, which is
simulated by an absorbing black sphere of radius a:
4a 
  r ( r  a )  G ( r , t )  c p ( r  a ,0)  dr 
0
t
2



2α 2 
τ 
2αe α

  erfc  α
 1 
,
1 τ 
1  τ 
πτ(1  τ) 


J (t )  4a 2 D p c p ( r, t ) |r  a 
 
1
 4D p a  3 / 2 3
 e 1 
 a p (1  ) 3 / 2 
2
c p ( r ,0) 
exp (  r 2 / a 2p )
π
3/ 2
a
3
p
G ( r, t ) 
,
exp (  r 2 / 4 D p t )
α
,
4πD p t
a
,
ap
τ
4Dpt
a 2p
(1)
,
(2)

where D p is the diffusion coefficient of the e out
, a 2p is the dispersion of e+ spatial (gaussian) distribution at
the end of its thermalization. Here we used standard representation for the solution of the radial diffusion
equation through the Green function of the 1-d diffusion equation:

1
c p ( r, t )   [G( r  a  s, t )  G( r  a  s, t )]( s  a )c p ( s  a,0)ds.
r0
Relationship (1) for J neglects the field dependent bias of the e+ distribution, c p ( r, t ) . Its accurate account
is difficult to be done analytically. So we do this approximately, modifying Eq.1 in the following way. The
factor [3 / 2a 3p (1  )3 / 2 ]1 represents e+ concentration in the center of the blob in the case of no e+ absorption. Let us substitute just this term by an expression, which accounts for the bias caused by the external
field:
a
1
2
p

 4Dpt 
3/ 2
 l p  b p Et 2 
 (   ) 2 
exp  2

exp 
1   
 a p  4 D p t 



,
3/ 2
3/ 2
π 3 / 2 a 3p 1  τ 
a 2p  4 D p t 

μ

eDa p
4T
, 
lp
ap
, E
D
. (3)

Here b p  eD p T is the mobility of thermalized e+ and l p is the drift of the e+ distribution during its thermalization:
tth
l p   υ p (t )dt 
0
eD(a 2p  abl2 ) 2 a 2p  abl2
eD tth
D
(
t
)
dt



,
p
T 0
6T
3
ap
(4)
tth
υ p ( t ) is the velocity of e+. Here the average e+ thermalization displacement squared, 6 D p (t )dt , during
0
its thermalization time, tth , was approximately equated to a 2p  abl2 . As far as a p is larger than the fieldinduced bias during the characteristic Ps formation time, approximation (3) is fine. PPs may be found from
the equation on the probability n inp of finding a positron inside the blob:
n inp  J (t )  e- 2 t  kep ne ninp / Vbl   2 ninp ,
Vbl  (2)3/ 2 abl3 ,
(5)
1

e (   ) - e (   ) 
n (0)  erf (  ) 
erf    
.
2 



2
in
p
2
(6)
 t
Here the factor e 2 accounts for e+ annihilation. In Eq.5 we assumed that all intrablob species ( e in , e  ,
ions) have the same spatial distribution  exp (  r 2 a bl2 ) and, therefore, the prescribed diffusion method
may be used for description of intrablob reactions [2,3]. The number of intrablob electrons, ne , decreases
394
Positron Annihilation - ICPA-12
due to ion-electron recombination as follows: ne (t )  n0 /(1   ei t ) , where  ei  n0kei / Vbl and  ei is the
respective rate constant. Integration of Eq.5 gives the following expression for PPs :
PPs  

k ep ne n inp
0
Vbl


0
0
dt  n inp (0)   J  e - 2 t dt   2  n inp dt.
(7)
We calculate the last integral term by approximation. Because  2 is the smallest parameter in the problem,

 2   ei  4Dp a 2p , and one can expect that  2  n inp dt  1 . From Eq.5 it follows
0
 t
0
 t kep ne
 np  e 2
n (t  )  n exp   
dt   2t  
,

 0 Vbl
 (1   ei t )
in
p
0
p

n 0p  n inp (0)   Je 2t dt,
0
γ
kep
kei
.
(8)
Substituting Eq.8 into Eq.7 one obtains
 t
 e 2 dt 



2
ei   2 ,  1
PPs  n  1   2 


 n 0p  1 
 n 0p .
(9)

0 (1   t )  
ei


  ei (   1) 
Dependence of PPs on D (or μ ) arises from n 0p only, through n inp (0) , Eq.6, and J, Eqs.2-3. In Eq.9 the
0
p
expression in the bracket is a prefactor, which does not depend on D.
Liquid
ap , A
a, A
C6F6
i-C8H18
n-C6H14
c-C6H12
C6H6
86(9)
221(6)
238(5)
292(7)
293(9)
68(7)
122(4)
124(3)
136(4)
176(6)
The results of the fitting to our previous data [2] based on the simplified relationship PPs  n 0p are shown in Fig.1 and Table. It is
worth mentioning that the present approach gives numerical estimation of the trapping radius, a, of the blob, i.e. the radius of the
shallow potential well, relevant to the e+-blob interaction. It is
about (2-3) abl ( abl is expected to be  40 Å). Differences of
a p between the present model and white blob model [2,3] are not
large, and give insight into real uncertainty of a model-dependent
parameter. As before, we have dPPs ( D) dD  0 at D  0 ,
which in the present approach is a consequence of e+ annihilation
competing with Ps formation.
Fig.1. Dependence of o-Ps intensity I 3  3PPs / 4  100% on external electric field D in liquids:  - cyclohexane (C6H12),  - nhexane (C6H14),  - isooctane (i-C8H18), ◊- hexafluorobenzene
(C6F6) and  - benzene (C6H6). Lines represent Eq.9,
PPs ( D)  n 0p , where n 0p is determined in Eq.8, Eq.6 and Eqs.2-3.
Values of b p ,  2 and  were taken from [2]. Values of adjustable parameters, a p and a , are shown in the Table.
Acknowledgements: This work is supported by the Russian Foundation of Basic Research (grant 00-0332918).
[1] Y. Ito, in Positron and Positronium Chemistry, edited by D. M. Schrader and Y. C. Jean, (Elsevier, Amsterdam, 1988), p.120.
[2] S.V. Stepanov, Cai-Lin Wang, Y. Kobayashi, V.M. Byakov, K. Hirata, Radiat. Phys. Chem., 58(5-6),
p.403, (2000).
[3] S.V. Stepanov, V.M. Byakov, J. Chem. Phys., (2002)
[4] A. Bisi, G. Gambarini, L. Zappa, Nuovo Cimento, 2D (5), 1465 (1983).