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 calculations, 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 following integrals:
2
0 x
2 dx,
∞
0 e
(−3r ) 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
an impact parameter of b = 3Å , find the value of the energy where an orbiting
trajectory occurs.
4. Deflection Angle: Write a program to calculate the classical deflection angle.
Use the same potential, scattering energies, and impact parameters as provided in
the previous problem. Do these angles correspond to the deflection angles from
your classical trajectory problem? Reproduce a plot similar to Fig. 1.7 that clearly
shows rainbow scattering and orbiting behavior.
5. Deflection Angle: Use the same potential as used in the previous two problems
to calculate classical action and the delay time. Provide a physical interpretation
for these two quantities.
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 calculations, 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 following integrals:
2
0 x
2 dx,
∞
0 e
(−3r ) 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
an impact parameter of b = 3Å , find the value of the energy where an orbiting
trajectory occurs.
4. Deflection Angle: Write a program to calculate the classical deflection angle.
Use the same potential, scattering energies, and impact parameters as provided in
the previous problem. Do these angles correspond to the deflection angles from
your classical trajectory problem? Reproduce a plot similar to Fig. 1.7 that clearly
shows rainbow scattering and orbiting behavior.
5. Deflection Angle: Use the same potential as used in the previous two problems
to calculate classical action and the delay time. Provide a physical interpretation
for these two quantities.