there is not an orbiting capture. There is instead a limiting deflection angle leading to
a small negative minimum that is usually called “rainbow” and offers a rationale for
some interference effects that will be commented later. As
b
increases (see Fig.
1.17
),
verylargecentrifugal(repulsive)contributionsalmostentirelyerasethepotentialwell
and make the deflection angle tend to zero.
1.5
Problems
1.5.1
Qualitative Problems
1.
Trajectories-Fixed Energy
: Without performing any calculations, describe the
qualitative behavior for the classical trajectories for a fixed energy and varying
the impact parameter from 0 to very large values. Provide a separate description
for each of the four given potentials.
2.
Deflection Angle-Fixed Energy
: Without performing any calculations, describe
the qualitative behavior for the deflection angle for a fixed energy and varying
the impact parameter from 0 to very large values. Provide a separate description
for each of the four given potentials.

1.5
Problems
37
3.
Trajectories-Fixed Impact Parameter
: Without performing any calculations,
describe the qualitative behavior for the classical trajectories for a fixed impact
parameter and varying the energy from 0 to very large values. Provide a separate
description for each of the four given potentials.
4.
Deflection Angle-Fixed Impact Parameter
: Without performing any calcula-
tions, describe the qualitative behavior for the deflection angle for a fixed impact
parameter and varying the energy from 0 to very large values. Provide a separate
description for each of the four given potentials.
1.5.2
Quantitative Problems
1.
Potentials
: The Lennard–Jones and the Morse potential qualitatively look similar.
The Lennard–Jones potential has two parameters ( ,
r
e
) or ( ,
σ
). How are
r
e
and
σ
related if the two forms of the potential are identical? The Morse potential has
three parameters ( ,
r
e
, and
β
). Derive an expression for
β
to make the Morse
potential have the same identical well depths
, equilibrium positions
r
e
, and the
value of
r
where they cross zero. Plot both potentials on the same graph and
explain the difference that you see.
2.
Numerical Integration
: Use the midpoint integration rule to integrate the fol-
lowing integrals:
2
0
x
2
dx
,
∞
0
e
(
−
3
r
)
dr
,
∞
σ
V
L J
(
r
)
dr
, and
∞
σ
V
Morse
(
r
)
dr
.
For the Lennard–Jones and Morse potential, use
=
140
.
9 kcal/mole, and
r
e
=
3
.
85Å
and choose
β
in the Morse potential so it crosses zero energy at
the same distance as the Lennard–Jones potential. Which potential form might
be better suited for scattering at very low energies?
3.
Trajectories
: Write a program to calculate trajectories for central field potentials.
When the potential and the impact parameter are both zero, does your program
produce correct results? Explain the trajectories you produce when the potential
is zero but the impact parameter
b
>
0. Now use the Lennard–Jones potential
with
=
140
.
9 kcal/mole and
r
e
=
3
.
85Å. Using your trajectory code to create
a table of deflection angles for 11 energies in the range
E
= [
20
,
120
]
kcal/mole
and 11 impact parameters in the range
b
= [
0
,
10
]
Å. Justify your results. For

#### You've reached the end of your free preview.

Want to read all 219 pages?

- Fall '19
- dr. ahmed