Rutherford Scattering

Transcription

1 Massachusetts Institute of Technology Physics Department rev-3 9/87 Junior Physics Laboratory: Experiment 15 Rutherford Scattering Scattering of Alpha Particles by Atomic Nuclei PREPARATORY QUESTIONS 1. What is the closest possible distance of approach of a 5.5 MeV a particle to a gold nucleus, and how does that distance compare to the actual size of the nuleus? 2. What is the definition of the differential scattering cross section? 3. How does die Rutherford cross section depend on the scattering angle, the atomic number of die scatterer, the target thickness, and the kinetic energy of the a particles? 4. Above what energy might the effective scattering cross section be essentially "geometrical", Le. equal to nr*a 2i3 m where r0 is the range of nuclear forces? 5. How do particles lose energy in traversing matter? 6. What are the effects of multiple scattering, energy loss, and finite sizes of sources and detectors on the accuracy and precision of the measurements? WHAT YOU WILL MEASURE 1. The intensity of a particles scattered by a thin metal foil as a function of angle for two metals of widely different atomic number ami two different a-particle energies. L INTRODUCTION Little was known about the structure of atoms when Geiger and Marsden began their experiments in 1909 at die Cavendish Laboratory on die scattering of a particles by thin metal films. A decade earlier at the Cavendish J. J. Thomson had discovered the electron and determined the ratio of its charge to mass by measuring the deflections of electron beams (cathode rays) by electric and magnetic fields. In 1909 Millikan measured Ac charge of the electron in the oil drop experiment Thus by 1909 both die charge and mass of the electron were known with considerable accuracy. Furthermore, Thomson's interpretation of X-ray scattering from carbon and other light dements had established that the number of electrons per atom of a given element was equal to its atomic number as determined by its position in die periodic table (not its atomic weight). Since the mass of an electron is much less than the mass of the lightest atom, hydrogen, it was clear that most of die mass in any atom is associated with the positive charge. The central problem was to figure

2 jgative parts lission spectra with die regularities expressed by die Balmer fonnula for hydrogt on rules and series limits for the complex spectra of multielectron atoms, iggling within die limitations erf Newtonian mechanics and Maxwell's electroma dieory Thomson worked with die idea that an atom is a sphere which the electrons occupy certain positions of equilibrium, like raisins in a pudding motion, the electrons should vibrate harmonically, radiating electromagnetic energy characteristic sharp frequencies that would be in die optical range if the radii of the atomic spheres were of the order of KT8 cm. However, die "raisin pudding" model yielded no explanation of the specific regularities of optical spectra. At this point Rutherford got the idea that the structure of atoms could be probed by observing the scattering of a particles, die positively charged emanation of radioactive substances that he had recently demonstrated were helium ions. According to die raisin pudding model an a particle traversing a thin gold film should suffer many small angle deflections as it passes close to or through die positive spheres of the gold atoms. Rutherford showed that die fraction of particles scattered in this way through an angle # or greater should decrease exponentially according to the equation where Qm is the mean multiple scattering angle. For a typical foil of gold leaf =30 one finds F^ of the order of exp(-30) or 10~13. Rutherford's formula turned out to be correct for very» small angles *^ of scattering. ^f Evidently» tiiere was substantial truth in the idea of multiple scattering. But in experiments initiated at Rutherford's direction, Geiger and Marsden (1909) found that 1 in 8000 a particles passing through a thin film of platinum was scattered through more dian 90! It was as though bullets fired at a bale of cotton could occasionally ricochet backward. At this point Rutherford (1911) advanced the hypothesis that die positive charge and most c die mass of an atom is concentrated in a "nucleus9* with dimensions of the order of 1 Orl 2 cm smaller than the atom as a whole with the electrons in some sort of confiuration ound it Using classical mechanics he calculated showed that die intensity of scattered particl (1) (f I A) (Ze/E^-fy 12), (2) where ^ is die scattering angle, Ze is the charge of die target nuclei, is the kinetic energy of the a particles, r is die thickness of the target (mass per unit area), and A is the atomic weight of die target

3 nuclei Further excruciating tedious these predictions. Gtiger hadn't invented the Griger oxmter yet, and electronic detection methods were still 20 years in die future. They used a low power microscope to observe and count by eye the scintillations produced by the a particles when they impinged on a screen lightly coated with zinc sulphide. To appreciate how hard that is, try doing with die luminiscent painted numbers on your wristwatch. Melissinos (1966) has given a thorough discussion of the Rutherford theory and the interpretation of data from a scattering experiment that is quite similar to dirt the exception of the specific detector and circuit arrangement Here we will assume you have studied that sectkm of his text and his discussion of solid-state detectors, and we will ccmfine our discussion to the features of the experimental setup and procedures that are peculiar to our setup. II. APPARATUS Figure 1 is a schematic drawing erf the apparatus under die bell jar. The source is which emits a particles of various discrete energies, die most frequent of which are MeV (86%), MeV (12.7%), and 5391 MeV (1.4%). All these decays lead to excited states of. The half-life of UlAm is 458 years. target ' a particle howitzer solid state detector r Figure 1. Schematic diagram of die apparatus for measuring the scattering of a particles. Note die difference between the howitzer position angle 0and the scattering angle #of a representative a-particlc trajectory. The source, deposited on a thin metal disk with die highest activity of 241 Am per unit area

4 tally available (-1.5 nullicuries per square inch) and sealed with an evaporated gold coating 1.5 microns thick, is covered by a metal washer with a 0.64 cm diameter hole and enclosed in a "howitzer" with a 0.64 cm diameter aperture in its snout Under vacuum a collimated beam of a particles emerges from the snout (the range of 5.8 MeV a particles in air at atmospheric pressure is only -4cm). The targets are mounted in a target holder which can hft manipulated from Outside die vacuum so as to bring a gold foil, titanium foil or no foil into the path between die source and the detector. The a particles are detected by a surface barrier silicon detector in die form of a 1.2 cm diameter cylindrical disk which acts as a solid state ionization chamber. A description of how such detectors work can be found in Melissinos (p. 208). CAUTIONARY NOTES DO NOT APPLY THE 100 VOLT BIAS VOLTAGE TO THE DETECTOR EXCEPT WHEN THE PRESSURE IS EITHER BELOW 100 MICRONS OR AT ATMOSPHERIC. AT INTERMEDIATE PRESSURES EVEN A SMALL VOLTAGE CAN CAUSE CORONA DISCHARGE WHICH WILL INDUCE VIOLENT CURRENT FLUCTUATIONS THAT CAN BLOW OUT THE SILICON DETECTOR AND/OR THE FET. ALSO, DO NOT CHANGE THE BIAS VOLTAGE SUDDENLY BUT TURN IT UP OR DOWN SLOWLY - NOT FASTER THAN 20 VOLTS PER SECOND. The solid-state detector and the preamplifier to which it is attached are delicate and must be treated with special care. The preamplifier has a special tow-noise field effect transistor which is easily ruined by an excessive current pulse. A sudden change in the voltage across the input capacitance will drive a large current through the FET. The manufacturer warns that the bias voltage on die detector must be turned off before the preamplifier is disconnected from die system. Too high a bias voltage will damage Ac surface barrier detector. The operating bias voltage on die detector is +100 volts (positive voltage oily). There is a voltage limiter in the bias supply line which should keep the voltage under die maximum allowable value of +150 volts. As die bias is first applied, look at die pulse output from the amplifier to be sure that breakdown is not occurring (the bias is applied across a very small gap, generally less than a few microns). The detector is lighi sensitive (die Schottkey barrio-acts like a photo-diode), and should be covered when die bias voltage is on. Turn down the bias voltage before exposing die detector to light The a-particle source is well-protected and secure (die radiation protection office examines it each year to be sure that it is completely sealed). It is not a particularly strong source (-100 ici or microcuries). Nevertheless, as widi any radioactive source, it must be treated with care. It is ^\

5 encased in die howitzer, and need never be disturbed The source should not be taken out of the apparatus or handled by anyone other than Tom White or a professor. ffl. PROCEDURE The goal of diis experiment is to observe the phenomena erf RudierfcMd scattering by measuring die dependence of die differential scattering cross section on die scattering angle, thez erf the target, and the energy of the a particles. An ideal set erf data from this experiment will consist of the following: (1) Measurements erf the peak channel positions and widths of the size distributions of the pulses produced by a particles that have traversed no foil, die gold foil, and die titanium foil with die howitzer position set at 0. (2) Measurements with no foil, the gold foil, and die titanium foil of the counting rates as a funtion of howitzer position angle out to the largest angles dial counting statistics and time limitations allow, for the full energy a particle beam. (3) Repeat of (2) widi the beam energy reduced by interposition of the gold foil energy reducer. (4) Measurements erf the relative positions of the howitzer, target and detector for each of die ital runs so that the geometrical factors involved in the data analysis can be calculated. (The analysis will be cleaner and simpler if precisely the same configuration is used in each erf the measurement runs). A* Explore die operation erf the equipment If the beu jar is under vacuum turn off the bias voltage, turn off the pump, and open the vacuum release valve. Lift die bell jar with die block and tackle, takin g care not to pull off die rubber gasket or bash the apparatus with the swinging bell jar. Find out how you can adjust die relative positions of the howitzer and detector with respect to the target, and how you can turn, raise and lower die target with the control rod that protrudes fiom the bottom of the chamber. Note particularly how you can s nultaneously rotate the howitzer and the target about a vertical axis through die target by turning die knurled outer cylinder under the chamber. This feature enables you to maintain a fixed relative orientation erf incident beam and target while you vary die range of scattering angles of die detected a B. Calibrate the measurement chain.

6 Move the howitzer support arm to 0 so that the ho witzer points directly at the detector through the empty target hole. Pump Ac system down with the bias voltage off. On the way down it is amusing to watch die pulses from the unbiased detector appear and gradually grow in j amplitude as die amount of air between die source and detector diminishes. When die pressure reaches 100 microns turn die bias voltage supply switch to die lowest setting (one click from "off") and turn the continuous bias voltage com Adjust the gain so that the a-particle pulses have an amplitude of about 5 volts. Then observe die pulse size distribution with die MCA and readjust the gains so that die peak of die a-particle pulse-size distribution lies at a convenient position, say channel 580. Connect the pulser to the test input of the detector preamplifier and adjust the pulser attenuator so that test pulses register somewhere in a channel above the peak of the a particles when the pulser dial control is cm Then measure the channel number of the lest pulse amplitude for various pulser dial settings over the range from 0.00 to on die dial. It is probably reasonable to assume that die amplitude of a detector pulse in the preamplifier at the point where the test pulses are introduced is proportional to die energy lost by the particle in die silicon device. Thus a plot of die channel number versus pulser amplitudes, scaled to match the energy of die a-particle pulses, should be an accurate calibration plot for the interpretation erf die pulse size distributions you will be measuring. C Measure die characteristics of the a-particle pulse-size distributions with no foil, the gold foil, and the titanium foil in place and with the howitzer at 0. fran^r ** tn» MQA ^cn? ^1^ a size spectrum of die a-particle pulses. With the cursors, measure die median channel number of the distribution, die positions of the half maximum rates on either side of the peak, and die positions of the edges of the distribution defined in some reasonable way, say die positions of the l/20th maximum rates. Place the start and stop cursors on either edge of the distribution, turn the intensify switch on, and record die number of pulses between the cursors and the accumulation time. Compare die results for no foil, the gold foil, and the titanium foil, and figure out die amount of energy lost by the a particles in traversing the films. With sufficient statistics in die no-foil run you may be able to resolve the two principal decay energies. D. Plan your scattering measurements. Exploit the territory by measuring the counting rates for various configurations of the howitzer, detector and targets. In particular, determine die effective angular profile of the beam by measuring the counting rate of pulses between the cursors on the MCA as a function of die angular position of the howitzer with no target foil. Then move the gold target foil into place and observe die effect erf Rutherford scattering on die counting rate at die smallest howitzer angle at which the counting rate without a fofl is zero. Then step back and plan how you will carry out the entire set

7 of scattering measurements to obtain the best possible set of data within the available time. Consider die following: The counting rates of scattered a particles over much erf the range of interesting howitzer angles are painfully small You will probably want to attain at least 10% statistical accuracy in each of your measured rates. If you occupy die first of die four lab sessions getting acquainted with the experiment, then you win have a total of about 9 hours in the next three sessions to get your scattering data plus a possible overnight run at a very large scattering angle to measure die wonderful phenomenon of atomic bullets ricocheting nearly straight back. Plan your measurements to make best use of the available time. For example, you may decide to measure the counting rates at every 5 setting of die howitzer position pointer, starting at 0. At small angles you might count for preset times of 100 seconds out to a howitzer angle where the accumulated count falls below some minimum acceptable number. For larger angles it might then be wise to measure die time required to accumulate a preset number of counts that will give you die minimum statistical accuracy you are willing to accept You can then proceed with the larger angle measurements until you run out erf the time you decide to allocate to that particular run. Then start over with the other foil or with the reduced energy beam. Note that you can reduce the incident energy by inserting into the slot in die howitzer die gold foil in the holder provided. You can increase die counting rates by sliding the detector up close to the target foil, but at the cost of broadening the range of scattering angles of die particles detected at any given howitzer angle setting and a consequent smearing of die results. In principle such smearing can be accounted for by an analysis involving a convolution of the predicted Rutherford angular dependence with die spread in scattering angle of the detected particles implied by the geometry of die howitzer, target and detector. But die broader die range of accepted angles die more important ami more uncertain will be die convolution calculation. You can also change the range of accepted scattering angles by changing the distance between die howitzer snout and die target Consider the possible virtues of keeping the plane of the target foils perpendicular to the incident beam at all times during your measurements at scattering angles less than 90 and positioning the howitzer snout as close as possible to the film. To witness alpha particles bouncing back from nearly head on collisions with gold nuclei (howitzer position angle -135 ) you wii' probably have to make an over-night run. At scattering angles greater than -60 you obviously have to back 17 with the howitzer and shoot from the haclrskle E. Measure die counting rate as a function of howitzer position angle 0for two incident energies of the a particles for no foil, the gold foil, and the titanium foil, preferably with precisely the same geometrical configuration.

8 8 V. DATA ANALYSIS AND INTERPRETATION ^ J I. Subtract the no-foil rates from die rates obtained with Ac metal foils in place to obtain the rates far scattered particles free of contaminfltinn hy th 2. Plot die subtracted rates versus sut^/z) and compare the trends with the prediction of the Rutherford theory for die variation of the differential cross section with sut*(#2). If you find a discrepency can you dunk of a possible explanation. You may find it wise to use 3- or 4-cycle loglog graph paper to handle the enormous range of counting rates and angle factors. 3. Detennine the energies lost by die a particles in traversing the foils and, with the aid of the table attached as Appendix A, deduce from those losses the thicknesses of die foils. 4. Allowing for the (r M) factors, compare the ratio of rates for the two different metal foils with die predicted Z dependence of the Rutherford formula. 5. Evaluate the differential cross sections for gold and titanium for die two different incident beam energies from die data obtained at a howitzer angle which is substantially larger than the angular width of the beam (to reduce die systematic error caused by die finite beam width), but where the statistical accuracy is still reasonably good. Compare die ratios erf the cross sections of gold at the two energies with die predictions of the Rutherford formula. 6. Explore the effects of die spread in scattering angle of the detected particles on the variation of die counting rate versus sut^fl/z) by convolving numerically die Rutherford angular factor with a function that approximately represents die angular spread. Can g(#,ft)j# the probability that a particle scattered at an angle between <f> and 4n-d<f> will be detected when the howitzer is at position angle ft Then we expect the counting rate at 0to be 77 (3) For a crude approximation one might represent ^ ly a triangular functicm defined by

9 = 0 (4) where 4b is the half- width at the bottom of the triangle. But note how this simplistic treatment ignores all the geometrical complications of evaluating die solid angle subtended by die detector at a given point of scattering in die foil and the range of angles of particles that can strike that given point 1. Gaiorowicz, S.,1974, Quantum Physics, Wiley, p Melissinos, A.C., 1966, Experiments in Modern Physcis, Academic Press, chapter Northcliffe and Schilling, 1970, "Range and Stopping-Power Tables for Heavy Nuclear Data Tables Part A, 7 4. Rutherford, E., 1911, "The Scattering of a and p Particles by Matter and the Sn the Atom", Philosophical Magazine, 21, Serge, E., 1977 Nuclei and Particles, Benjamin, chapter 2.