Geoscience Reference
In-Depth Information
The quantities
i
.
v
/ and .
v
/
D
i
.
v
/
C
ˇ
.
v
/ are the rate of ionization and the
total decay determined by the t expressions,
i
.
v
/
D
t
vv
1
,and
ˇ
.
v
/
D
V
ˇ
v
.
D
˝
ˇ
ˇ
ˇ
V
ˇ
v
.R/
j
v
i
V
ˇ
v
is equal to
V
ˇ
v
D
V
ˇ
.R
e
/˛
ˇ
v
,
The matrix element
where ˛
ˇ
v
D
˝
ˇ
ˇ
ˇ
v
i
(the index
v
0
refers to the Rydberg channels open with
the vibrational quantum number
v
0
v
1). Substituting Eq.
1.31
in the general
expression (Eq.
1.30
) and summing over
j
for the partial cross sections, one has
4
2
R
e
V
ˇ
E
C
E
ˇ
.ˇ/
AI
.E/
D
g
ˇ
˛
ˇ
v
0
"
v
0
C
˛
ˇ
v
0
C1
.
v
0
C
1/
."
v
0
ı
v
0
;
v
1
C
!.1
ı
v
0
;
v
1
//
:
(1.32)
v
1
X
i
.
v
0
C
1/
v
0
D
0
Here "
v
D
E
E
v
is the electron energy in the
v
-th motion channel, and ı
v
;
v
0
is the Kronecker symbol. Note that all the parameters in Eq.
1.32
are the adiabatic
system parameters that can be calculated in the framework of the existing theoretical
methods or reconstructed from the absorption spectra.
Near the threshold, the total cross section of the AI reaction is a linear function
of the energy (E<!):
AI
.E/
D
E;
(1.33)
where the quantity is equal to
˛
ˇ0
C
˛
ˇ1
D
4
2
R
e
X
ˇ
g
ˇ
V
ˇ
E
C
E
ˇ
i
.1/
.1/
That is, the threshold behavior in the quantum case is absolutely different from the
law
E
3/2
, resulting in the semiclassical theory (Smirnov
1981
; Devdariani
1979
).
This result has a simple explanation, because the transition matrix elements in the
presence of the Coulomb field do not vanish at
E
D
0. Therefore, in the region
0 <
E
<¨the cross section is proportional to the phase space angular momentum
j
max
, corresponding to the given value of the
E
energy, i.e., linearly with
energy. Moreover, if Eq.
1.32
is to exclude the communication with the Rydberg
states (putting
1
D
0) and to ignore the tunneling effects and calculate the Franck-
Condon factors using the Landau-Zener approximation for their estimation, then
!M
1=2
c
.2"
0
/
1=2
F
.c/
ˇ
˛
ˇ
.E/
D
;
