Rutherford Orbits

In this computational lab you will explore the trajectories of particles in scattering experiments where a beam of atomic nuclei are incident on a stationary target. The scattering angle  θ  relative to the incident beam direction is closely related to the trajectory of the projectile when it interacts with the target. That trajectory reveals the nature of the force acting between the beam and the target and can be used to measure the size of atomic nuclei. The figure below shows the trajectory of a (^6)Li nucleus as it passes near a  (^208)Pb nucleus. The incoming (^6)Li starts out moving parallel to the z-axis at a distance b (called the impact parameter), gets repelled by the Coulomb force in the vicinity of the target, and ends up moving away at an angle (Subscript[θ, s]). In Section 1 are commands you might find useful for this lab. In Section 2 you use the orbit equation to calculate and study the trajectory of the scattered beam.

1. Necessary Commands

It may be useful in this laboratory to make plots of functions in polar coordinates. Some examples of how to do that with the PolarPlot command are shown below. You may also want to show multiple graphs on a single frame. Use the Show command demonstrated here.

p1 = PolarPlot[0.5,{x,0,2*π}]
p2= PolarPlot[0.5+ 0.05*Sin[10*(x)],{x,0,2*π},PlotStyle->RGBColor[1,0,0]]

Show[p1,p2]

You may also need to make semi-log plots of the measured (^6)Li + (^208)Pb  Rutherford cross section with and without uncertainties. Here is an example of using the Table command to construct a set of points that we plot.

t1 = Table[{t,10*Exp[-0.25*t]},{t,0,12,2.0}]; (* table from 0 to 12 deg in steps of 2.0. *)
p3=ListLogPlot[t1,
Frame->True,FrameLabel->{"t (s)", "N"},PlotStyle->{RGBColor[0,1,0],PointSize[0.02]},
BaseStyle->FontSize->12
]

Here is  the same data now with uncertainties.

(* create a random table of uncertainties. *)
unc = Table[t1[[i,2]]*0.5,{i,1,Length[t1]}];
(* add the uncertainties to the cross sections in a new table. *)
t2=Table[{t1[[i,1]],Around[t1[[i,2]],unc[[i]]]},{i,1,Length[t1]}];
p4=ListLogPlot[t2]

2. Understanding the orbit equation

We showed in the class the orbit equation that governs central force motion is

r(θ) = ((L^2)/(μα (1 + ϵ cos(θ - (Subscript[θ, 0])))))

where

α=-(Subscript[k, e])(Subscript[q, 1])(Subscript[q, 2])         ϵ = (Sqrt[1 + ((2E (L^2))/((μα)^2))])             L = (Subscript[μv, 0])b      (Subscript[θ, 0])= -ArcCos[(1/ϵ)]

μ is the reduced mass, b is the impact parameter, and (Subscript[v, 0]) is the initial speed of the projectile.  In this section you will investigate how the different constants in r(θ) effect the trajectory of the incident particle.

1. Set the values of the constants in Eq. 1-2 to reasonable values. Below are some suggestions. Generate a few trajectories using the PolarPlot command demonstrated in Part 1. Be aware that you may have to adjust the angular range of the plot to eliminate jumps in the Rutherford trajectory.

impact=1.0; (* impact parameter in fm *)
hbarc = 197;  (* ℏc *)
energy = 73.7; (* MeV *)
mLi = 6*931.5; mPb = 208*931.5; (* mass in MeV/(c^2) *)
ZLi = 3; ZPb=82; (* no. of charges *)
Lvalue= impact*mLi*v0; (* angular momentum in fm-MeV/c*)
alpha =-ZLi*ZPb*hbarc/137;  (* Coulomb constant *)

2. For a fixed value of the energy E=73.7 MeV  vary the impact parameter b. How does the distance of closest approach (DOCA) of the (^6)Li to the (^208)Pb change with impact parameter? How does the trajectory change? The lead nucleus has a radius of about 7.1 fm.

3. Now fix the impact parameter b at a small value and vary the incident energy E of the beam particle. Do calculations for at least one energy greater than the original value of E=73.7 MeV and at least one smaller. What happens to the trajectory of the (^6)Li nucleus? How close does the (^6)Li get to the (^208)Pb  as you change the energy? How does the DOCA depend on energy for a given impact parameter?

4. For what impact parameter b will the DOCA be a minimum? Why? What is the scattering angle of the (^6)Li for this impact parameter?  Generate a formula for the DOCA for (^6)Li +(^208)Pb for this impact parameter.  Apply it to each of your beam energies. Is your calculation consistent with your plots?