Introuction Alpha Particle scattering Revise Jan. 11, 014 In this lab you will stuy the interaction of α-particles ( 4 He) with matter, in particular energy loss an elastic scattering from a gol target etection, or Rutherfor scattering. The concepts explore in this experiment have several applications in nuclear physics an nuclear meicine. Energy Loss by Charge Particles by Scattering As charge particles traverse matter, they lose energy by ionizing the atoms along their path. As the amount of energy lost per collision is approximately constant (at least for small energy losses), each particle will have a path length (or range) in a given stoppen material epenent on its incient energy. So for any particular type of particle the range is a efinite function of the energy (as illustrate in Figure 1) Figure 1. -particle range curve.
Energy loss by collision is itself statistical in nature an the number of collisions is huge in macroscopic ranges. The number of collisions varies with small fluctuations from an average, an so the stopping power, or average energy loss per unit path length E/x, can be calculate for charge incoming particles as a function of materials an incoming energies. The rate of energy loss of alfa-particles is given (Melissinos) by: 4 E 4Z e me 1 n eln x m. (1) I e Here Z 1 e is the charge of the incoming particle, is its velocity (non-relativistic), an n is the e electron ensity of the scattering material n e Z N A, () A (where is the ensity of the material in g/cm 3, N is Avogaro's number, Z an A A are the atomic an mass number of scattering material). I is an average ionization potential of this material. An estimation for I is: 3 I Z 9.1 1 1. 8Z ev. (3) Differential cross-section The main quantity which can be etermine from any colliing experiment is the ifferential cross-section. Its magnitute characterizes the probability of scattering as a function of scattering angle. Suppose our etector is a ifferential element of soli angle, an there is an angle between the etector an incient beam of alpha-particles. If I is the flow ensity of the beam, an if n is the number of scattering centers (i.e. number of particles in that volume of target which is occupie by beam) then the number of particles scattere in the ifferential element of soli angle per unit time is N I n, (4) where is the ifferential cross-section: N I n. (5) This quantity, which has units of area, epens on angle characterizes the probability of scattering into a given angle. is proportional to the ifferential element of soli angle, occupie by the etector, so in orer to avoi ifferent meanings in various experiments, physicists usually measure the ifferential cross section per unit of soli angle element:
N, I n. (6) To calculate this, we nee to measure two values: I an N, an we also nee to etermine the number of scattering centers: n 0 n LS (7) ( n - concentration of atoms in target, L - target thickness, S - area of beam's cross-section, 0 which is efine by the collimator aperture), an soli angle element: S R (8) ( S - area of etector, R - istance between target an etector). Rutherfor Scattering In 1909 Geiger an Marsen scattere alpha-particles from nuclei an iscovere that there were consierable numbers of large angle scatterings. Much more than expecte, though of course fewer than at small angles. In Rutherfor scattering experiment in 1911, alpha particles from a raioactive source were allowe to strike a thin gol foil. Alpha particles prouce a tiny, but visible flash of light when they were striking a fluorescent screen (Figure ). Figure. Rutherfor scattering apparatus. Surprisingly, alpha particles were foun at large eflection angles an some were even foun to be back-scattere. This experiment showe that the positive matter in atoms is concentrate in an increibly small volume. Thus the raisin puing moel of atoms was estroye, an Rutherfor experiment gave birth to the iea of the atomic nucleus.
The thickness L of the target in the Rutherfor scattering experiment is of great significance. It must be "thin". This means that mean free path of alpha-particles must be much greater than L : L. This requirement guarantees that the alpha-particle will collie with the nuclei in the target only once. The value of the ifferential cross section oes not epen on the number of scattering centers (it is calculate per scattering center ), so we can consier this quantity in the case of scattering the alpha-particle by only one scattering center. A significant role in the theory relies on the choice of the scattering potential U r. This value characterizes the properties of the target an oes not epen on experiment conitions. Rutherfor chose a Coulomb's potential Ze Ur. (9) r As a target, we use a Au nucleus which has a mass much greater than an alpha-particle's mass, so we can neglect the energy loss ue to recoil of the target nucleus. Calculations of alpha-particle trajectories in Coulomb's fiel show that they are hyperbolas, an trajectories iffer by impact parameter b (istance from axis to alpha-particle when it is far from nucleus, see Fig.3), which epens on angle : Z Z e E ctg 1 b, (10) here Z 1 e is alpha-particle charge ( Z ), Z e is charge of target particle, E - energy of alpha-particle. 1 Figure 3. Scheme of scattering alpha-particle by nucleus.
Let's rewrite the ifferential cross-section for the case of scattering by one center as here N N N I n I N n, (11) is the number of scattere alpha-particles per unit time by one center. Such representation is convenient to connect the measurable macroscopical parameter scattering angle - with the non-measurable microscopic impact parameter b. As shown in Fig.3, before the particles passes through the area S, they pass through an element b b of a ring, which is a istance b from the axis (scattering center is situate on this axis). The number of particles which pass through this element of area per unit time is N Then. (1) Ib b N b b I. (13) After integration over from 0 to, will represent the area of the ring which is in the left part of Fig.. After using the formula given above for the non-measurable parameter b, we will erive the famous scattering ifferential cross-section formula which carries Rutherfor's name: 1 e Z1Z 4 m (here S R alpha-particle, an 1 4 sin sin, R is istance to the area S ; e Z e is the charge of target nucleus). (14) Z 1 is the charge of Analysis of the hunre thousan or more scattering events recore for the alphas on gol fully confirme the angular epenence preicte by Rutherfor. Apparatus The apparatus consists of a basic vacuum system, a bell jar, a fixe raioactive source holer/collimator, a rotatable soli state etector an associate amplifier an pulse height analyzer. The charge particle etector is a Si-ioe (you can fin a goo escription of the functioning of such evices in Melissinos an Napolitano). The etector is a back-biase ioe which allows electron an hole carrier particles to be prouce an subsequently collecte in a
so-calle eplete region. Roughly 3 ev are require per charge. So a 1 MeV particle has a charge 10 6 Q e. On a 00 pf capacitor this will prouce a 0.5 mv signal. 3 There are two raioactive sources; they are both curium winow. The target is a thin gol (Au) foil. 44 Cm. The weaker one, has a thin Proceure BEFORE STARTING: The etector requires about 50-70 V to work properly. Be sure that the jar bell is covere by the black light shiel before voltage is appllie. Energy loss measurements 1. Place the thin winow source close to the etector. Seal the bell jar an light shiel. Evacuate the bell jar. Measure the spectrum seen in the etector as a function of applie voltage. For this, start the bias voltage at 10-0 V an turn up slowly, making sure not to excee the maximum allowe for the etector (70 V). Plot the peak position an with as a function of applie voltage.. Measure the position an with of the peak as a function of pressure in the bell jar. Thin winow source shoul be place about 5 cm from etector. To o this, bring the bell jar up to atmospheric pressure, turn off the voltage an then remove the shieling an bell jar. After moving thin winow source, re-seal the bell jar an light shiel an apply the voltage. Plot the position an with of the peak as a function of pressure; compare your results to theory. Recor the count rate for ifferent pressures (use Region-of-Interest (ROI) button). 3. Bring the bell jar up to atmospheric pressure. Turn off the voltage. Remove the shieling an bell jar. Remove the thin winow sourse. Place etector so it points at the collimate sourse. Re-seal the bell jar an shiel. Apply the etector voltage again an etermine the peak position an with. Why is it shifte? Rutherfor Scattering 1. Remove voltage. Open the bell jar an place the gol target in place. Reclose the bell jar an re-evacuate the jar. Apply voltage. Measure the peak position. How much energy is lost in the gol? Can you thus etermine the thickness of the gol foil that is use as target? You might want to use this information: - alpha-particle energy in Au: 6.8 MeV
E MeV cm - 01.5 x g. Measure the integrate peak rate an position as a function of angle. More points in the forwar irection, but out as far as possible. You will nee to increase the time of observation for larger angles. 3. Plot the peak rate an position as a function of angle an compare to theory. To o this you will nee to know the geometry of the system: make appropriate measurements of the istances an apertures an inclue them in your report.