Rutherford Scattering

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All pages in this lab. Note To print Full Lab Write-up click on each link below and print separately


I. Rutherford Scattering

II. RUT Pre Lab Questions and Staff Sign Off Sheets.....PRINT, FILL THIS OUT, Turn it in


III. Error Analysis Notes

Reprints and other information can be found on the Physics 111 Library Site




Contents

Before The Lab

View the 'Radiation Safety Video' From Inside the 111-Lab, Paste into your Browser [ U:\Advanced Lab Share\Safety Manuals and Videos ] and then click on Rad Safety Video located in the My Computer directory on the 111-Lab Network Share Drive, get Radiation Safety form from GSI, then fill it out & sign the Radiation pink form, and get a Radiation Ring.


Discuss the Physics about this experiment with the faculty or the GSI's in the 111-Lab before starting.

You should keep a laboratory notebook. The notebook should contain a detailed record of everything that was done and how/why it was done, as well as all of the data and analysis, also with plenty of how/why entries. This will aid you when you write your report.


View the Rutherford Scattering video, discuss the pre-lab questions with an instructor, and have the Staff Sign-Off Sheet (RUT) signed.

Prerequisite Reading Materials

Watch the video first! Then read reference 1.

  1. A. C. Melissinos, Experiments in Modern Physics, 2nd Edition, Academic Press, New York, 2003.
  2. E. Rutherford," The Scattering of α and β Particles by Matter and the Structures of the Atom, "Philosophical Magazine 21, 699(1911). This article is conveniently reprinted in Foundations of Nuclear Physics, R.T. Beyer compiler., Dover Publications, New York, 1949. This reference appeals to students who are attracted to the historical aspects of the experiment. It contains references to other papers of historical importance.
  3. R. D. Evans, The Atomic Nucleus, McGraw Hill (1972).
  4. "Rutherford Experiment", Physics 111-Lab Reprints in Physics Library.
  5. Comfort, J.R., et al "Energy Loss and Straggling of Alpha Particles in Metal Foils"; Physical Review: Vol. 150, No. 1, Oct. 1966, pp. 249-256.
  6. "High-Energy Particle Data Volume II"; UC Radiation Lab: UCRL 2426, two pages.
  7. Goulding, F.S., & Landis, D. "Linear Amplifier Gating & Timing System"; Instrumentation Techniques in Nuclear Pulse Analysis, No. 1184, pp. 121-133.
  8. At the lab station there is an Americium 241 Data Sheet and a manual for the Tracor 7200 Pulse Height Analyzer. Please take the time to read these.

A collection of reprints that includes a good many references is available from the Physics 111 library. There are many texts which incorporate a discussion of Rutherford Scattering at the junior level, such as Kennard, et al., Modern Physics.

Introduction

The Rutherford Scattering Experiment, in which α particles are scattered by a gold foil, is one of the most famous experiments ever performed in physics, because it demonstrated the validity of the nuclear model for the atom and permitted a direct measurement of the nuclear charge. When alpha particles in a beam strike a gold foil and collide with individual gold nuclei, they are elastically scattered (no loss of energy). The collisions and the angles into which the the alpha particles are scattered are described by a model in which both the gold nuclei and the alpha particles act like charged spherical particles. The force of repulsion between them is described by Coulomb's Law. The fraction of particles that is scattered into a particular solid angle at a given direction to the incoming beam is called the differential cross section and is given by \frac{d\sigma}{d\Omega}=\left (\frac{Z_1Z_2e^2}{4E} \right )^2\sin^{-4}{\frac{\theta}{2}}

where particle 1 is the helium nucleus (alpha particle), particle 2 is the gold nucleus, and E is the kinetic energy of the alpha particle. Of course there are electrons around each gold nucleus, but they are so light that the energetic alpha particles push them aside with a relatively small loss of energy.

In the experiment you will measure the relative numbers of particles scattered as a function of scattering angle. You will observe the \sin^{-4}{\frac{\theta}{2}} dependence and from your data calculate the nuclear charge of gold .

Theory

It is your responsibility to reproduce the necessary derivations for this experiment. Information on Rutherford scattering is usually first presented in Physics 7C. Detailed derivations are given in Physics 137A or Physics 105 courses and their associated texts. The recommended text for this course (Melissinos) is an excellent source for Rutherford Scattering. The bottom line is that you will need to find a relationship between what you measure - number of counts in a given time interval as a function of scattering angle - to the charge on the nucleus. The Rutherford formula and some algebra give Z=K\sqrt{N}\sin^2{\frac{\theta}{2}}

where K is a constant depending upon the foil thickness, the strength of the source, the angular beam-width \triangle\theta and the counting time. N is the number of counts.

There are some models that are helpful in thinking about this experiment. The Rutherford scattering derivation assumes a single heavy charged spherical target and an α particle. But this experiment uses a gold foil, not single gold nuclei. What difference does this make to your experiment? It may help to think of the possible effects by breaking down the problem into two views. First, imagine the gold foil as an array of immobile gold atoms in space. Now remove the electrons from your picture. The remaining array of fixed bare nuclei is the first order experiment of Rutherford scattering. In this view you can see what it means to have a single scattering event or multiple scattering interactions.

To think of other effects in scattering, imagine the same gold foil array but remove the nuclei from your picture. What remains is a combination of fixed localized electrons (bound electrons) and an electron cloud (conduction electrons) in space. This picture greatly complicates the interpretation of the experimental data. α particles lose some energy in passing through this array and cloud of electrons. Because the electron-α particle interaction is actually quite complicated and difficult to treat, we create approximations for the interaction.

In our discussions we assumed that alpha particles have definite energies that you look up in a table. They do when emitted, but when we use them they do not, because the source is enclosed in a metallic capsule and the particles lose energy in passing through the capsule wall. Melissinos has a discussion on the loss of energy by alpha particles passing through matter. It is also true that they lose energy as they pass through the foil, as mentioned above. You must account for this when you analyze your data. The energy loss changes as the scattering angle changes. There is simply not enough time to go into detailed determinations of the energy loss. Instead, take the alpha particle energy for this experiment to be on average 3.77 MeV, and use this in your calculation of the nuclear charge of gold. You may wonder how Rutherford coped with this and other problems, since he had no sophisticated equipment. He had very much stronger radioactive sources, the sources were not shielded (the shields are a major source of energy loss), and he had people rather than a solid state detector and electronics to count scintillations, the flashes of light emitted when alpha particles strike a fluorescent screen used as a detector.

There are other reactions taking place, that we ignore: Cherenkov radiation, bremsstrahlung and inelastic nuclear interactions. If you have the time and interest, you can pursue the subject of energy losses in either of the two references given immediately below.

  • J.R. Comfort, et al., "Energy Loss and Straggling of α Particles in Metal Foils, "Phys. Rev. 150, 249 (1966). Includes data for gold foils.
  • J.H. Atkinson, Jr. and B.H. Wills, "High Energy Data, Volume II," UCRL Report No. 2426 (1957). Page 35 of this report is a useful Range-Energy plot for α-particles of 4 to 10 MeV Kinetic energy. Gold is not included, but lead is.

Experimental Overview

The overall plan of the experiment is as follows. Alpha particles from a radioactive source called an alpha gun are made into a beam by two collimating apertures and directed toward a thin gold foil. Particles scattered by gold nuclei are counted by a detector placed at various angles to the beam direction. A histogram of particle counts vs. energy is made at each angle chosen. Even though nuclear scattering is elastic, some energy is always lost because of interactions with extra-nuclear electrons as mentioned above, and these losses are not exactly the same for every alpha particle. Consequently the histogram looks like a broad line rather than a delta function. The area under the line is proportional to the number of counts. With these numbers the angular scattering law mentioned above can be demonstrated, and the nuclear charge calculated.

The source of alpha particles is the isotope 241 of Americium. Its half-life is 458 years. The energies of emitted alpha particles are 5.49 MeV (86%), 5.44 MeV (13%), and 5.39 MeV (1%). The Americium is deposited as a thin layer on an aluminum foil located in the gun as shown below in Fig. 1.

Figure 1

Diagram of Vacuum Chamber

The alpha particle gun and foil are in held in fixed positions when data are being taken, but the detector position can be moved with respect to the foil while at a constant distance from it. The particle beam is collimated by two apertures adjustable from the alpha gun. If necessary, changing the aperture separation is the easiest way to modify the collimation, since it can be changed without losing vacuum. It is also possible to remove the gun from the chamber and the removable aperture changed in size or removed entirely, but do not touch the interior aperture. This aperture size is already measured and can be found in Fig. 3.

If we use a well collimated beam from the alpha-gun in order to have a well defined angular resolution, the total α-particle flux is weak so that counting rates at any appreciable scattering angle are extremely low. You will need a few all-night and week-end data collection runs. Plan accordingly.

The experiment is done in a vacuum chamber because the range of alpha particles in air at atmospheric pressure is only a few cm.

Procedure: getting started on Rutherford Scattering

You must have a radiation ring and wear it when working near the apparatus, since there is an Americium-241 source which emits alpha particles and gamma rays. For safety, check the apparatus with a Geiger counter before you begin (counter usually kept at X-Ray or Gamma Ray experiments).

  1. Familiarize yourself with the apparatus. Refer to Figs. 1, 2, 3. Start by removing the front cover from the chamber. Put the screws somewhere where they won't be lost. Open the chamber and look at its construction. The detector is an extremely fragile device. There is a microscopically minute platinum wire between the attachment post and the evaporated gold surface of the detector. Break anything and you'll be out of commission for a week, minimum!
    Figure 2: Diagram of apparatus. Top View
  2. Gently change the position of the detector using the "detector angle adjusting screw" shown in Fig. 2. Leave it at about zero degrees (protractor on front face of chamber). Remove the foil holder from the chamber by grasping it firmly, pulling and wiggling it from side to side.
    Fig. 3: Alpha Gun
    Remove the second aperture of the collimator, the one flush with the end of the gun, with the following procedure: loosen the brass rectangular clamp located at the end of the source tube on the brass rod (far right) of the alpha gun. Slowly push the rod into the tube-the second collimating aperture should fall out. Leave the rod positioned such that the first collimator is flush with the end of the source tube. Now loosen the brass "O-ring" clamp (nut) on the source tube, move the tube to the center of the chamber, and then retighten the clamp (finger tight). Also move the hose clamp on the source tube so that it is flush against the brass O-ring clamp, and tighten the hose clamp. It is important to retighten this clamp in the proper position after moving the gun. Otherwise atmospheric pressure on the outside will push the gun into the detector arm when the chamber is evacuated. Now make a run with no foil, non-collimated, point-blank, in order to find the beam flux, energy intensity profile and proper electronic settings.
  3. Pump down the chamber with the following procedure. Making sure that the cover is clean replace the chamber cover and tighten the thumb screws by hand only. Check to see if the cover is evenly seated on the vacuum chamber. Open the intermediate venting valve between the chamber and the pump. Start the pump you then should here a hissing sound when the pump is running. The switch is located near the motor. Slowly close the valve the sound should go away. \[To open the chamber, reverse the process-slowly open the valve 1st and listen for the hissing sound, then turn off the pump\]. These precautions prevent shock waves from damaging the delicate gold foils, and minimize the amount of oil vapor from the pump entering the chamber. We are now going to use the scope to follow the signal through the electronics to set everything at proper levels. Do this carefully and ask questions if something does not seem right, but don't expect an Instructor to do your thinking for you and tell you what to do.
    Figure 4 (aka Figure 2 immediately above)
    Figure 4 is a block diagram which gives some idea of the electronic logic. Use Fig. 5 to follow the details of making adjustments.
    Figure 5
  4. Turn the electronics on. There are four switches to check: the master power switch in the lower right side of the rack, the LAS (linear amplifier system) power switch on (light does not work) the middle right side of the rack, the multiple outlet power switch behind the rack, the "intensity"/power switch below the screen of the PHA, and the detector power at the top of the rack.
  5. DETECTOR and PRE-AMP: The detector is a Silcon PN-junction with about 250um of gold on it. It has a bias of 50 volts at 4 microamps with a diameter of 0.75 inches. The detector receives it power through a voltage divider to get 50 volts from the 100 volt power supply in the rack. Take the output from the pre-amp (BNC coaxial cable) and feed it into the scope. Adjust the A TRIGGER LEVEL on the scope and see if you can obtain the positive pulse as shown in Fig. 5. You may have to increase the intensity of the trace, and put a hood on the oscilloscope to shut out room light. This pulse is the amplified signal from the detector. The positive voltage pulses represent the detected alpha particles, and the height of the pulses is proportional to the energy of the particles. Reconnect the output of the pre-amp to the AMP IN of the linear amplifier (PG1).
  6. LINEAR AMPLIFIER: This unit amplifies and shapes the unipolar pulse (unipolar means it goes up and comes down without crossing zero) from the pre-amp. The polarity knob must be set properly for the input pulses, so we want it set to POSITIVE. To start with, also set the AMP GAIN knob to NORMAL, COARSE GAIN to 50, INTEGRATOR to 0.5 microseconds, and DIFFERENTIATOR to 0.5 microseconds. For now, set the FINE GAIN between 2.5 and 3. Use the plug-in probe to look at point PG2 ---the output of the Linear Amplifier---on the front of the Linear Amplifier System Chassis (upper left corner). You should see a signal like Fig. 5, only it will jitter. You may have to adjust again the A TRIGGER LEVEL on the scope, and make sure that you're triggering on a positive slope.
  7. BIPHASE SHAPER: this unit converts the pulse from unipolar to bipolar (a positive pulse followed by a negative pulse). Use the plug-in probe to look at point PG3, the output of the biphase shaper. You should see something similar to Fig. 5.
  8. DELAY AMPLIFIER/SLOW COINCIDENCE/LINEAR GATE: The Delay Amplifier takes the same signal going into the SCA and delivers it without significant change in amplitude, but delayed, to the linear gate. The delay is necessary for the functioning of the linear gate because of the time required for decision making in the SCA, which controls the opening of the linear gate. The Slow Coincidence unit controls the operation of the linear gate. For our purposes we will use it to take the SCA in and out of the circuit. Set the EXT and FAST COINCIDENCE switches to OUT (since these inputs are not used in the circuit), and for now set the SCA switch to OUT as well. If you now probe at PG6 -- the output of the Linear Gate-you should see roughly the same thing as you saw at PG3.
  9. SINGLE CHANNEL ANALYZER: The SCA looks at a pulse, and checks to see whether its height lies within limits that you set. If it does, then it outputs a logic pulse as shown, and this pulse opens the linear gate and allows the signal to pass through. If the pulse height is not within the limits that you set, then the gate does not open and the pulse does not go on to the MXA. This allows you to screen out unwanted signals. For now, set the Upper Level to 10 and the Lower Level to 0, to pass everything. The output can be seen at PG11 and should look like Fig. 5. Note that you need to match impedances here: use a BNC "tee" with a 50 ohm terminator on one arm and the output from PG11 on the other arm, with the stem going into the scope.
  10. BIASED AMPLIFIER: Not used. Set GAIN knob to OUT.
  11. OUTPUT SHAPER: This circuit stretches the positive part of the pulse and attenuates the negative part, to make the pulse more suitable for the MCA. It does not alter the height of the pulse. You may connect the scope to PG16 to see the output of the shaper (and hence the input to the MCA). Compare with Fig. A5c. You may, if you wish, leave this cable connected for the remainder of the experiment. You should also know that input signals to the MCA must lie between 0 and 7.5 volts-check out your pulse height to be sure it falls in this range.
  12. We are now going to set the gain so our signal is appropriate for the PHA, and we will set the SCA windows. At this point you must be familiar with the operation of the TRACOR PHA. Read the manual! The relevant sections are marked; refer especially to Section 13.00. To set up the PHA (Pulse Height Analyzer), select memory group 1/1. Begin accumulating counts by pressing ACQUIRE. Adjust the FINE GAIN on the Linear Amplifier to place the energy peak in the upper 1/4 of the PHA channels (watch on the scope as you make changes in gain). Note that you want to keep the Dead Time at less than 10%. To do this you may need to adjust the Lower Level Discriminator (LLD) to keep the MCA from counting low level noise. Adjust no higher than 0.20 LLD.
  13. There is a LabView Program ICON on the computer desktop called "Tracor 7200 PHA". This program automates the data taking. Just double click on the Icon and take data. The program will Start Acquire, Stop Acquire, Clear PHA memory, Download the data from the PHA to the computer and save it on the computer. You will need to make a folder in "C:\User" whose file name is no more than 8 letters long. The program is a DOS program that works verys well but does not accept longer names. Now start the PHA via the program and take data for about 5 minutes. You should see the histogram appearing on the computer and the PHA. With the computer, stop data takintg and download the data to the computerl This is a test. You need to save your data file as something like "data2.dat", that can be read by any spreadsheet program for plotting, such as Analyze, Excel, Axium, KaleidakGraph.
  14. To set the SCA window, "select" the area under the peak on the PHA, leaving a little room to both sides of the peak. (Sec. 9 of the TRACOR manual; remember that the manual was written by people who are familiar with the instrument and think like its designers, so don't give up if the language is opaque or obscure. Persevere, and you may figure it out. If not, ask). Set the SCA switch to IN on the SLOW COINCIDENCE panel to put the SCA in use, and watch the effects of varying the upper and lower level controls of the SCA. The MODE switch should be in the NORMAL position. You want to set the upper and lower levels so that you don't lose any of your signal, but you don't want to have any unwanted signals either. Consider cutting out your signal, and then "backing off" until you have left only that part that you want. You should also leave a little room for drift in the electronics, perhaps half a turn of the upper level knob. Leave all electronic settings fixed for the remainder of the experiment. Most PHA settings can be changed at will, with no effects on the data.

Later on if there are problems in observing the scattering peak on the PHA, adjustments may need to be made to the SCA until it is selecting just the pulses which define the peak observed in step 2. To do this, gate the PHA with the output pulse from the SCA. Make a preliminary setting of the window by observing counts on the scalar. These counts on the scalar should be proportional to the counts on the PHA when all is set up correctly. Gating of the SCA is visible on the screen of the PHA. The proper selection of this gating is part of the noise elimination scheme. It is useful to graph two different spectrums, with and without gating on the same sheet so as to note precisely where the SCA window cuts into the spectrum. Finalize the settings for the gating window and record them for future reference. Note that the high energy side of the peak has a much steeper rise than the low energy side. For lack of a better name this is called the Landau distribution. Be sure to leave a little margin in the gating for drift in the electronics. The amplifier settings should, of course, never be changed from this point on in the experiment.

Handling gold foils

Since the scattering target is a gold foil, you need to know something about them.' 'The foils are like swiss cheese so you use double layers of gold leaf foil, each layer of which is about 10 milligrams by weight because of the holes and very fragile. Each pair of foils is fastened to a numbered holder by plastic tape. A weighed sample of each foil sheet is kept with the experiment. The weights should be recorded in the log book for each holder used. With them you can determine the surface density of each foil. In some cases this surface density is tabulated directly in the log book.

Handle the foils by the edges of the brass holder. Place foils carefully in the slotted repository when they are not in the chamber. The retaining fingers of the foil holder in the chamber are purposely loose so as not to abrade the tape. Keep them that way .

You are now READY TO GO ! ! !

Data Collection (see Appendix on Error Analysis)

Error Anlaysis Notes [1]

Careful planning is essential if you want to complete this experiment. You can't possibly take all the data you need for a research-type project. As you take data, you will see where you have to make compromises. Talk them over with an instructor at every stage.

First you need to decide what you are going to measure. To do this, write down the equations you are going to use, identify those quantities you can measure (such as angle of scattering), those that are given (like the charge of the electron), and those that you want to determine. Then decide how you are going to make the measurements and what parameters you are going to vary to make the measurement an optimum for accuracy and time.

For example, you need to measure scattering angles. Therefore you need to establish the zero-degree position of the detector. How are you going to do this? Can you do it without the gold foil in place? Now is a good time to talk with an instructor, to be sure you are on the right track. Don't ask first - figure it out for yourself first, then explain to the instructor what you are going to do and why.

You will need to determine solid angles of the detector. You may want to change the solid angle for some measurements, by changing the aperture size.

The signals get weaker as the angles get larger. Think ahead, about how much time each data-taking run will require.

You may find it helpful to make a few trial runs, and analyze the data from them, before taking final measurements. If you are not doing the right thing, you want to find out about it immediately.

Finally, analyze your data, remembering that your goal is to observe the angular dependence of the Rutherford scattering formula, and to calculate the atomic number of gold.

One way to test the angular dependence of the differential scattering cross section is to fit a straight line to the curve of log(count rate) versus log of the fourth power of the sine of the half angle, and to determine the slope of this line by a linear least square fit. Follow the example given in Lyons, Data Analysis for Physical Science Students, Chapter 2, especially Section 2.9, p. 63ff, in which is a worked example.

Questions

  1. At what scattering angles in your experiment will deviations from the Rutherford scattering law occur because of non-zero size of the nucleus?
  2. At what scattering angles will deviations occur because of screening of the nucleus by electrons?
  3. To what extent (be quantitative) do you expect alpha particles to experience multiple scattering in the foil?
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