2013 Purdue University 1

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1 Lab #11 Collisions: Rutherford Scattering OBJECTIVES In this lab you will: Use the energy and momentum principles to analyze the problem of Rutherford scattering, and estimate the initial conditions to be used in the VPython program. Complete the VPython program that implements Rutherford scattering, and analyze the plots of the momentum of the particles produced. Observe how the scattering angle varies with the impact parameter. 1. Analyzing the System & Completing the Program 1.1 Estimating Initial Conditions Collisions are the main tool for probing atomic and sub-atomic particles. A classic experiment by Lord Rutherford used fairly high energy alpha particles to probe how the protons were distributed in the atom. He guessed that the protons were all located in a tiny fraction of the volume of the atom. His experiment confirmed that the protons were all located in such a small volume (now known as the nucleus). We will use a computer model to re-create his experiment. This computer modeling of the experiment by Lord Rutherford is described in detail in your textbook (Problem 10.P.38). From Blackboard Learn, open the program rutherford.py in this week s Lab folder. It will use the momentum principle to test various trajectories of the alpha particles as they collide with a gold nucleus. It simplifies the problem by reducing it to two dimensions (no initial z-components of the position or momentum for either particle). The gold nucleus, which has 79 protons (charge of 79 x 1.6x10-19 C) and 197 total nucleons (mass of 197 x 1.7x10-27 kg) is initially located at the origin. The alpha particle, which has 2 protons (charge of 2 x 1.6x10-19 C) and 4 total nucleons (mass of 4 x 1.7x10-27 kg) is initially located somewhere on the x-axis (you ll choose an appropriate location). The initial momentum of the alpha particle is in the positive x-direction. You will calculate this momentum given that the alpha particles have a kinetic energy of 10MeV (10x10 6 ev) when they are far from the nucleus (recall that 1.6x10-19 J = one ev). Answer the following questions: 1.) Using the energy principle, find how close the alpha particle gets to the gold nucleus in a head-on collision. Use the initial kinetic energies given above. You may assume that the alpha particle and the gold nucleus start very far apart. (The gold atom s electrons may be ignored since they are much lighter than either of these particles and are easily scattered away from the nucleus by the approaching alpha particle with little loss of energy to the alpha particle only a few ev loss.) 2013 Purdue University 1

2 We now want to choose an initial position for the alpha particle for our program. If we start the alpha particle too far away, the program will take too long to complete (since it will take a great while for the alpha particle to get near the gold nucleus). If we start the particle too close to the nucleus, then our energy calculation in part 1 is no longer accurate (since we are not starting the particle far enough away). We need to choose an initial position so that the strength of the interaction between the particle and the nucleus ( the coulomb force ) is much smaller than it will be when the two are very close. 2.) Choose the initial position of the alpha particle so that the force on the alpha from the gold nucleus is 100 times smaller than it will be at closest approach. Remember that the gold nucleus is at the origin, and that the alpha particle is sitting somewhere on the negative x-axis. Use the space below to calculate this position symbolically. 3.) Calculate the force on the alpha particle at the closest approach distance (symbolically). 4.) Calculate the initial momentum of the alpha particle (again, symbolically) Purdue University 2

3 5.) We want to choose a Δt such that magnitude of the change in momentum ( p) of the alpha particle at closest approach is much smaller than the magnitude of the initial momentum of the alpha particle. Using the force you calculated in question 3, pick a Δt such that p is 100 times smaller than the initial momentum. Show your work in determining this time step in the space below. 6.) What is the initial velocity of the alpha particle? Is this a relativistic problem? How can you tell? 7.) Calculate the length of the while loop. Take the total length of the alpha particle s trajectory (~ twice the magnitude of the initial position) and divide it by the initial velocity. This gives a good estimate of the total time needed. Why might this be so? 8.) In the experiment, the alpha particles have an initial momentum in the positive x-direction, but they may travel along or parallel to the x-axis. The y-offset of their initial position is known as the impact parameter b. Sketch what the paths of the alpha particle and gold nucleus should look like with a nonzero impact parameter, so that you can recognize whether the program is behaving sensibly Purdue University 3

4 CHECKPOINT 1: Raise your hand and ask your instructor to check your work. You may continue on while you wait. 1.2 Completing the Program You now need to add in the constants you have just calculated into the Rutherford program, and complete the missing lines of code. Fill in a reasonable value for the impact parameter (b). Keep in mind the distance scales in which you are working! Input the initial position of the alpha particle. Input the masses (alpha.m and gold.m) and charges (alpha.c and gold.c) of the alpha particle and gold nucleus. Input the initial kinetic energy (alpha.k) of the alpha particle. Input the intial momentum (alpha.p) of the alpha particle, in terms of the initial kinetic energy of the alpha particle. Input the Δt (delta_t) the program should use. Input the length of time that the simulation is to run (simtime). Complete all of the lines inside the loop, which calculate the net force on the alpha particle (and the net force on the gold nucleus) as well as updating the momentum and position of the alpha particle and the gold nucleus. Run your program and see if it is behaving as you expect. 2. Plotting the p x and p y for the alpha particle and the gold nucleus Sketch what the plots of p x and p y for the alpha particle and for the gold nucleus should look like as a function of time, so that you can recognize whether the program is behaving sensibly. You should draw a curve for each particle on the graphs below. Assume a nonzero impact parameter. DO THIS BEFORE LOOKING AT THE PLOTS FROM YOUR PROGRAM! p x p y t t 2013 Purdue University 4

5 Before your while loop, you need to create graph windows and graphing curve objects for displaying the x components of momentum for the alpha particle and the gold nucleus, as well as the y components of momentum for both particles. The necessary statements are already in your program, so you just need to remove the comment symbols from the lines indicated below. #Graphing Objects gdisplay(xtitle='time',ytitle='px', x=400, y=0, width=400,height=200) alphapx=gcurve(color=color.blue) goldpx=gcurve(color=color.yellow) totalpx=gcurve(color=color.green) # this is for the px of gold + alpha gdisplay(xtitle='time',ytitle='py', x=400, y=200, width=400,height=200) alphapy=gcurve(color=color.blue) goldpy=gcurve(color=color.yellow) totalpy=gcurve(color=color.green) # this is for the py of gold + alpha Inside the while loop, after updating the time, uncomment the lines of code which add a point to the gcurve objects for each value of t. Run your program and use the plots to answer the questions below. 9.) What can be said about the total momenta of the alpha particle-gold particle system? Use the momentum principle to support your observation. 10.) How are the y-momenta of the alpha particle and the gold nucleus both before the collision related? What about after the collision? Use the momentum principle to support your observation. CHECKPOINT 2: Raise your hand and ask your instructor to check your work. You may continue on while you wait Purdue University 5

6 3. The Impact Parameter The impact parameter b is the magnitude of the initial y-coordinate of the alpha particle. If b=0, the collision with the gold nucleus is head-on and the scattering angle is 180 degrees (backscattered). The scattering angle decreases as b increases. Answer the following questions. 11.) Write the scattering angle (θ) in terms of the x and y components of the final momentum of the alpha particle p α,f,x and p α,f,y (after the collision). Uncomment the last 3 lines of your VPython code, which will calculate and print the scattering angle in terms of the x and y components of the final momentum of the alpha particle. The impact parameter is also printed. Try a few different impact parameters to see the different trajectories and scattering angles that they produce. Answer the following questions by trying out various impact parameters. 12.) What impact parameter gives approximately a scattering angle of 170 degrees? 13.) What impact parameter gives approximately a scattering angle of 10 degrees? 14.) If you wanted to scatter alpha particles at an angle of 170 degrees or greater, what range of impact parameters are allowed? Compute the size (area) of the target that the alpha particle has to hit so that it scatters at an angle of 170 degrees or greater (this is called a cross-section ). Figure on page 397 of your text will be helpful in visualizing the target Purdue University 6

7 15.) If you wanted to scatter alpha particles at an angle of 10 degrees or smaller, what range of impact parameters are allowed? (Remember we have found from our ball-spring model of matter that aluminum atoms are spaced by roughly 2x10-10 m). Compute the size (area) of the target that the alpha particle has to hit so that it scatters at an angle of 10 degrees or less (this is called a cross-section ). Figure on page 397 of your text will be helpful in visualizing the target. If you were firing alpha particles at a gold foil it is nearly impossible to know the impact parameter of each particle. Instead, you send many of them in a uniform beam so that a large region of the foil is illuminated. By comparing the relative number of alpha particles scattered at various angles, we can understand the structure of the atom. 16.) Compare the areas around an atom that will give scattering angles of less than 10 degrees and of more than 170 degrees. Which is more likely to occur? 17.) What is the ratio of the number of scattered alpha particles that are scattered by less than 10 degrees to the number scattered by more than 170 degrees (i.e. the ratio of the areas you found above)? 18.) Imagine you were Lord Rutherford and you did the experiment and found that the ratio of the number of alpha particles that are scattered less than 10 degrees to the number scattered by more than 170 degrees was approximately 1. What would that tell you about your model of the atom? Would you change it in any way? CHECKPOINT 3: Raise your hand and ask your instructor to check your work Purdue University 7

1 Newton s 2nd and 3rd Laws

Physics 13 - Winter 2007 Lab 2 Instructions 1 Newton s 2nd and 3rd Laws 1. Work through the tutorial called Newton s Second and Third Laws on pages 31-34 in the UW Tutorials in Introductory Physics workbook.