Rutherford Scattering

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Mabel Newman
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1 Rutherford Scattering Today's understanding of the atom, as a structure whose positive charge and majority of mass are concentrated in a minute nucleus, is due to the α particle scattering experiments conducted by Ernest Rutherford and his colleagues ( ). The essential features of Rutherford's apparatus are shown in Figure 1: α particles emitted from a radioactive source strike a thin gold foil. Most pass straight through the foil, but a fraction are scattered at an angle θ into the detector. The smaller the distance of closest approach between an α particle and a gold nucleus, the larger is the scattering angle. Vacuum chamber Collimators 41 Am! Source Detector " Source and target foil are mounted on a commong support and can rotate Au foil holder Figure 1: Setup for Rutherford Scattering

2 Background Information A theoretical analysis of the scattering process under the assumption of the existence of a small massive nucleus leads to the following expression for the cross section: ( ) d! d" = zze 4#$ 0 1 ( ) ( 4E kin ) ( ) sin 4 % (1.1) where z is the charge of the projectile (for an α particle z = ) and Z is the charge of the nucleus (for Au Z = 79), E kin is the kinetic energy of the projectile (for 41 Am the α particle has an energy of MeV) and! is the scattering angle. The quantity d! is the differential cross section and d! is the solid angle. The geometric d" interpretation of the cross section and solid angle are shown in Figure cross section = are of ring of radius b and width db R d! b! d! Particles hitting the ring between b and b+db are scattered by an angle between and Solid angle of the entire ring :!! + d! They are scattered into a larger ring on a sphere with the scattering nucleus in its center! Rsin(")Rd" d! = =! sin(")d" solid angle of R small area: Figure : Geometry of the cross section and the solid angle d! = d!rsin(")rd" = sin(")d"d! R The cross section given in equation (1.1) is for one nucleus only. In order to calculate the rate at which particles are scattered at a certain angle one needs to

3 ! N know the flux of the incoming particles j =! inc $ " # A % &, the number of nuclei in the target per unit area and finally one needs to determine the solid angle of the detector. The number of target nuclei per unit area is given by t T!N a Mmol where t T is the target thickness,! is the density of the target material, M mol the atomic mass and N a Avogadro s number. The solid angle for small detectors openings is defined as!" = A det R where A det is the active detector area and R is the distance between the target and the detector. The observed rate therefore is!n! = t T "N a M mol A det R ( zze ) ( 4#$ 0 ) 4E kin ( ) 1 ( ) sin 4! (1.) For a given target the observed rate is therefore of the form: C!N! = # (! "! sin 4 0 )& $ % ' ( The goal of this experiment is to check where this behavior is observed and to determine the constants C and! 0. (1.3) Experimental Procedure Your equipment consists of a vacuum chamber with a rotatable source and target mount and a semi conductor detector. The chamber is connected to a vacuum pump. As a target you use a gold foil of µm thickness. This foil is very fragile be very careful and do not touch it! There are two slits that need to be installed between the foil and the source which define the size of the target spot and determine the average flux of incoming alphas. The detector is connected to a pre- amplifier, then to an amplifier and to a multi channel analyzer (MCA) that you have encountered previously. On the cover of the vacuum chamber is a scale that indicates the angle between the beam of α particles and the detector (the angle! ). Without a target set the angle to 0. Pumping When pumping or venting the vacuum chamber you should always use the same procedure: Place the target and source combination at 0 degree. This protects the target foil from damage by the air stream in or out of the chamber The little brass valve must be closed when you turn the pump on or off

4 For evacuating: close the valve, connect the hose to the pump. Turn on the pump. Very slowly open the valve and let the air be pumped out of the chamber. You will hear the air flow and the sound of the pump change. This should take about 0 s. Now you are ready to take data For venting: close the valve. Turn off the pump. Make sure the valve is closed. Disconnect the hose from the pump. Very slowly open the valve and let the air stream back into the chamber. This should also take about 0 s to detector to source slit foil holder Notches for alignment Figure 3: Slit and target holder Notches for alignment Taking data 1. Set the MCA live time to 300s and take a spectrum without target. You should see a peak, corresponding to the α particles, which are mono energetic. The width of the peak is due to the thickness of the source itself where the α particles loose energy and the energy resolution of the detector.. Install the gold target with the 1mm slit (see figure Figure 3). Make sure that the notches fit into their counter parts in the target holder. The large circle needs to face the detector and the slit faces the source. 3. Close the vacuum chamber, make sure the target position is at 0 degrees and pump down. 4. Take another spectrum. Note how the peak has shifted. This is due to the energy loss of the α particles in the target. 5. Set the live time to 600s (Aquire! MCB Properties! Presets) and take a spectrum at 0, 15, 10, 0, - 10, - 15, and - 0 degrees. Take also spectra at - 5 and 5 degrees with a live time of 00s 6. Set the live time to 100s for - 30 degrees and 1800s for - 40 degrees Remember : save all the spectra as SPE (ASCII) files. In that way you can analyze them later using the LabTools package.

5 Figure 4 shows an example spectrum with the gold foil at 0 degrees. Analysis Figure 4: Au spectrum at 0 degrees Determine the count rates (counts/time) for each angle. In order to check if you observe indeed Rutherford scattering calculate the logarithm of the count rate (and its error) and plot this versus the logarithm of sin(! / ). You should see a linear relation ship. Make sure you take the absolute value of!. You will most likely see that the rates left and right for the same angle are different. This is due to a possible offset in your angle measurement. Try to add or subtract! o and see if the results improves. Determine above which angle you probably see a linear relationship. For those angles fit a line and determine the slope. Does is agree with the behavior of Rutherford scattering? In order to accurately determine the angle offset you will determine the coefficients in equation (1.3), namely C and! 0, via a non- linear fit of the experimental count rates. Then make a semi- log plot of the count rate as a function of! and plot the fitted curve. Discuss your observations and results. Analysis Hints How to sum bins in a histogram For each spectrum add the counts in the peak. As this is a simple spectrum with only one peak, you can basically just add all channels (or bins) above a certain value. For the example in Figure 4, you could add the channels between 400 and Assuming the spectrum is called sp when you load it, you get this sum with the command : C, dc = sp.sum(400, 1000) # sum the channels between 400 and 1000

6 C, dc = sp.sum(00) # sum all channels above 00 Here C is the sum and dc is the uncertainty in the sum. How to get the live time of a spectrum The live time is stored in the title of the spectrum. The function below allows you to extract the number from the title. # get the live time from a spectrum def get_time(sp): tt = float(sp.title.split('=')[1].split('s')[0]) return tt # Put this in your analysis script and you can get the time by doing: t = get_time(sp) R = C/t dr = dc/t # get the time # calculate the rate # calculate the error in the rate How to do the non- linear fit In order to determine the parameters of the angular distribution you need to define the function and its parameters. This is done as follows (please check the LabTools documentation for more explanations): C = B.Parameter(10., 'C') # define the parameter C t0 = B.Parameter(0., 'theta_0') # define parameter for! 0 def S(x): # define the function return C()/B.np.sin(0.5*(x - t0()) )**4 With these definitions you are ready to carry out the fit: sfit = B.genfit( S, [C, t0], \ x = theta_r,\ y = R,\ y_err = dr) Where the fit results are stored in sfit, theta_r is the scattering angle in radians, R the experimental rates and dr the uncertainties. Make sure that these arrays contain only those values that you want to use in the fit. How to do a semi- log plot of data and fit First plot the data as a semi- log plot: B.plot_exp(theta_r, R, dr, logy=true)

7 Then plot the fit: B.plot_line(sfit.xpl, sfit.ypl, color = 'g') B.pl.show() Make sure that theta and sfit.xpl have the same units

Atomic and Nuclear Physics

Atomic and Nuclear Physics Nuclear physics Rutherford scattering LD Physics Leaflets P6.5.2.1 Rutherford scattering: measuring the scattering rate as a function of the scattering angle and the atomic number