Use numerical integration techniques to explore possible trajectories of an alpha particle when it collides with a gold nucleus at different impact parameters. Your program should display the paths of the alpha particle and of the gold nucleus, and also plot the x and y components of momentum of both particles as a function of time. We will consider motion only in two dimensions (Figure 10.44; a slice through a 3-D beam of alpha particles). You may find it useful to work through Problem 10.P.32 as background for this computational model.
An alpha particle is a helium nucleus (two protons and two neutrons, charge +2e); in the Rutherford experiment its initial
kinetic energy was about 10 MeV (remember that 1 eV = 1.6 × 10−19 J). The gold nucleus is inside a piece of gold foil and
is initially stationary; its charge is +79e and it contains 197 nucleons. Initially we will ignore the interactions of the alpha
particle with the cloud of electrons surrounding the nucleus, which are in fact small.
(a) In this model it is important to pick reasonable initial conditions and display scales—otherwise you may not see
anything meaningful. Before beginning your program, calculate or estimate reasonable values for the following and
report your results:
▪ Closest approach. For a head-on collision, calculate approximately the distance of closest approach, given that
the alpha particle and the gold nucleus repel each other. Assume that the gold atom's electrons are scattered
away by the incoming alpha particle.
▪ Initial position of the alpha particle. Given the distance of closest approach, what would be a reasonable
initial x for the alpha particle? What would be a good range to explore for the impact parameter b (initial y
coordinate of the alpha particle)? Use these estimates to decide on a scale for the coordinates used to display
the trajectories.
▪ Value of Δt. Trial and error alone will not be very helpful in estimating Δt. You can calculate the initial speed
of the alpha particle, and you know the approximate size scale for the collision. Choose a value of Δt such
that the alpha particle doesn't move very far in one step.
▪ Time scale. To plot px and py vs. t during the interaction, you must select a scale for the time axis that allows
you to observe the entire interaction. About how long should the time axis be?
▪ Momentum scale. What should the maximum value of the px axis be? You should scale the py axis to the
same value, to make comparisons easy.
▪ Exit criterion. You will need to exit from your computational loop when the particles have moved far apart.
How will you test for this?
(b) Write a program that displays:
▪ The paths of the alpha particle and the gold nucleus.
▪ The value of the impact parameter b, and the value of the corresponding scattering angle (angle between the
initial path of the alpha particle and its final path).
The angle can be calculated by taking the arctangent of py/px, using the final values.
▪ A plot of px for each particle and total px, and a similar plot for py. (Consider what these plots should look
like if the program is working properly.) Report what value of Δt gives adequate accuracy. What is your
criterion?
Report values of the impact parameter b that produce the following:
• A very small scattering angle
• An angle around 90 degrees
• A large deflection angle (back-scattering)
(c) What other physical quantity could you plot that would provide a check on the accuracy of your calculation?
Carry out this check and report what value of Δt provides good accuracy according to this check.
(d) To see what a different model of matter would have predicted, replace the gold nucleus in your model by a proton.
Describe briefly what you observe. (Note that if the protons were uniformly spread out throughout the volume of the
atom, they would be far enough apart that on average an alpha particle would interact with only one of them.)