Chemistry Reference
In-Depth Information
q
ν
0
r
b
L
φ
FIGURE 3.16
Polarization interaction between the charged particle (point charge
q
)and
polarized neutral particle in the center of the laboratory system. At the impact parameter
b
≤
b
L
the charged particle is captured in the attractive potential.
α
E
grad
E
.
F
pol
=
p
ind
·
grad
E
=
·
(3.95)
F
r
·
dr
α
dE
dr
dr
ε
pot
=−
=−
·
E
(
r
)
·
(3.96)
α
α
E
2
=−
·
E
·
dE
=−
2
.
α
SI
·
q
2
ε
pot
(
r
)
=
ε
pol
(
r
)
=−
r
4
.
(3.97)
2
·
(
4πε
0
)
·
2
The scattering of the charged particle in the polarization potential is described by the
conservation of the mechanical energy (3.98) and the angular momentum (3.99) in
the laboratory system for polar coordinates, see Figure 3.16. The elimination of the
angular velocity and the condition of vanishing radial velocity at the minimal radial
distance
r
0
results in a quadratic equation (3.100) for the determination of
r
0
.
dr
dt
2
2
r
2
d
φ
dt
m
2
·
m
2
·
α
SI
·
q
2
ε
=
v
0
=
+
−
r
4
.
(3.98)
2
·
(
4πε
0
)
·
2
d
φ
dt
→
d
φ
dt
=
b
v
0
r
2
·
·
v
0
·
=
·
r
2
·
m
b
m
.
(3.99)
A real solution of (3.100) exists for the case that the impact parameter satisfies the
relation (3.101).
r
0
2
α
SI
·
q
2
−
b
2
·
r
0
+
m
=
0.
(3.100)
2
(
4πε
0
)
·
v
0
·
b
4
4
−
α
SI
·
q
2
m
≥
0.
(3.101)
(
4πε
0
)
2
·
v
0
·
Solving (3.101) for the minimum impact parameter
b
min
=
b
L
and taking into account
the relative velocity
v
r
of the particles and the reduced mass
m
red
in the center of mass
