There is instead a limiting deflection angle leading to a small negative

There is instead a limiting deflection angle leading

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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.
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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
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