How Big Is a Nucleus?

In this computational lab you will explore some of the properties of scattering experiments where a beam of sub-atomic particles is 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. The figure below shows the differential cross section data from of a (^6)Li- (^208)Pb scattering experiment. In Section 1 are commands you might find useful for this lab. In Section 2 you compare the calculated Rutherford scattering cross section with measurements of (^6)Li + (^208)Pb  at (Subscript[E, Li])= 73.7 MeV and extract the size of the lead nucleus.

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 go 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. Measuring the size of the lead nucleus.

The differential cross section can be used to measure the size of the nucleus. For elastic scattering the Rutherford cross is

((dσ)/(dΩ)) = ((((((Subscript[Z, 1])(Subscript[Z, 2])(e^2))/(4(Subscript[E, cm])))))^2) (1/((sin^4)((θ/2))))

where  (Subscript[Z, 1])and (Subscript[Z, 2]) are the number of charges in each nucleus, (Subscript[E, cm]) is the center-of-mass energy of the beam, and θ  is the scattering angle of the beam. This is a classical calculation and serves as a prediction for the measurement. The properties of the measured (^6)Li + (^208)Pb  cross section and this prediction of the Rutherford cross section can be used along the results of the previous section to measure the size of the nucleus. 

1. First obtain the data for the  (^6)Li + (^208)Pb  cross section at (Subscript[E, lab]) = 73.7 MeV from the course website at the address below.

https://facultystaff.richmond.edu/~ggilfoyl/cm/notebooks/LiLeadMeasurements737.nb

2. Generate a plot of the data including the uncertainties. Calculate the Rutherford cross section for  (^6)Li + (^208)Pb   and compare it with your data. You may find a semi-log plot to be useful. Clearly show your calculations. Does the Rutherford cross section provide a reasonable description of the data? Where does it succeed? Where does it fail?

3. What is the size of the gold nucleus?