Environmental Engineering Reference
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where the exchange interaction potential is
Δ
( R )
D ε
g ( R )
ε
u ( R ). For the cross
section
σ
res of resonant charge exchange this gives
Z
Z
1
1
Δ
( R )
2
sin 2
σ
D
2
π
d
(
),
(
)
D
dt .
(2.42)
res
1
0
From this expression, the average probability of resonant charge exchange is 1/2 for
small impact parameters
, so the cross section of this process may be represented
in the form
D 2 R 0 , w e
1
v
σ
Δ
( R 0 )
,
(2.43)
res
where
is the relative collision velocity. Because at large R we have
Δ
( R )
v
2 J / m e e 4 ) 1/2 ( J is the ionization potential), we obtain
the following velocity dependence for the cross section of resonant charge ex-
change [55, 56]
exp(
γ
R ), and
γ D
(2
ln 2 C
v
,
D 2
σ
res ( v )
γ
2
and this dependence may be represented in the form
R 0
2
D 2
1
γ
ln v 0
v
σ
res ( v )
C
,
(2.44)
R 0 /2isthecrosssectionofresonantchargeexchangeattherelativeve-
locity v . In slow collisions R 0
where
π
1, and Table 2.6 contains the values of this
parameter along with the cross sections of resonant charge exchange at an ion en-
ergy of 1 eV in the laboratory frame of reference (an atom is motionless) [57-59].
These values were calculated on the basis of the asymptotic theory [60-63], which
involves expansion of the cross section over a small parameter 1/( R 0
γ
). With the
exceptionofatomsandionsofthefirstandsecondgroupsoftheperiodictable,
the resonant charge exchange process is entangled with the processes of rotation
of angular and spin momenta, and the cross sections in Table 2.5 are averaged
over initial momentum directions. The accuracy of the averaged cross sections in
Table 2.5 is several percent. If this cross section is approximated by the velocity
dependence
γ
res ( v 0 ) v 0
v
σ
res ( v )
D σ
,
(2.45)
we have in this limiting case
2
D
R 0
1.
γ
Because of the approximation character of (2.45), the accuracy of this relation be-
tween parameters
and R 0
is restricted. Table 2.6 contains values of this param-
eter for a collision energy of 1 eV.
γ
 
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