Geoscience Reference
In-Depth Information
Fig. 3.13
Angular
momentum L
2
.r/ versus ion
distance
r
from the center. At
positive energies,
E
> 0, ions
with L<L.a/can reach the
particle surface
a
R
0.0
r
0.0
DISTANCE FROM CENTER
L
2
<2mR
2
.
j
E
j C j
U.R/
j
/ will
cross
the
particle
surface
(see
Fig.
3.14
).
Otherwise the restriction L
2
<2ma
2
.
j
E
j C j
U.a/
j
/ works. Hence,
L
a
D
min
L
2
.a/;L
2
.R/
for the attractive nonsingular potentials. The notation L
a
stands for the angular
momentum separating the trajectories crossing the particle surface from those
passing aside.
It is easy to find that L
a
D
L
2
.R/ in the interval of energies (
E
< 0):
R
2
j
U.R/
j
a
2
j
U.a/
j
j
U.R/
j
<
j
E
j
<E
0
D
:
R
2
a
2
In the case of the Coulomb attraction, this interval is narrow:
e
2
R
<
j
E
j
<
e
2
R
C
a
