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Rutherford Scattering Advanced Lab: Rutherford Scattering Conner Herndon and Nick Lucas (Dated: 21 April 2015) Leading up to the 20th century, experiments pointed toward the existence of fundamental constituents known as atoms. As the atomic theory developed, the plum pudding model stood strong. By shining α-particles on gold foil, we replicate Rutherford’s famous experiment with modern technology. We predict and observe a 1/ sin4 (θ/2) trend with increasing detection angle. This result is in direct opposition to the plum pudding model and implies a dense, positive core at the center of the atom. I. BACKGROUND John Dalton, a famous chemist and physicist from the early 19th century, proposed the existence of a fundamental entity in an effort to describe contemporary chemistry1 . Dalton argued that if identical chemicals were formed as aggregates of dissimilar particles, there would be inconsistent specific gravities of materials. For this reason Dalton proposed that fundamental particles known as “atoms” of the same element hold identical mass, size, and chemical properties. Although John Dalton captured the fundamental concept of modern atomic theory through his philosophical and chemical treatment, experimental proof did not arrive for nearly a century. This theory of atoms resounded strongly with chemists, however, physicists disregarded atomic theory until the early 20th century, despite Ludwig Boltzmann’s statistical approach to thermodynamics which assumed atomic theory2 . With Jean Baptiste Perrin’s experiment3 and Albert Einstein’s theoretical explanation of Brownian motion4 , atomic theory finally gained a foothold in physics. Even with the existence of atoms, many phenomena may not be adequately explained by their mere existence such as emission spectra. As stated by Gottfried and Yan, “[...] if the atoms are structureless, how can they display an excitation spectrum – can a structureless piano play a sonata?”5 Before the acceptance of the atom, a more fundamental particle – the electron – had been experimentally discovered and studied by J. J. Thomson in 1897 (as detailed by the authors in a previous report6 ). Understanding the electron in terms of this newly discovered particle became the forefront goal of the physical theory of chemistry. J. J. Thomson offered his plum pudding model which features a positive electric “fluid” within which electrons oscillate7 . According to Thomson’s model, when the electronic “plums” were disturbed by gas collisions or electromagnetic radiation, they would oscillate wildly within the “pudding” and emit corresponding frequencies of light. Unfortunately, the model did not accurately predict the frequencies observed. At the turn of the 20th century, a young Ernest Rutherford and his colleague Paul Villard began work in radioactive decay in an attempt to understand the struc- FIG. 1. Rutherford experiment apparatus. Americium (source) shines α-particles at the gold foil. The scattered particles are then detected at an angle θ from the direct path. ture of the atom.8 By 1907 Rutherford and Thomas Royds determined that α-radiation is made up of fast moving helium ions.9 While performing experiments with α-particle bombardment on very thin gold foil, Rutherford noticed extreme deflections of the α-particle beam. Rutherford counted the number of scattered α-particles versus scattering angle through a microscope,10,11 and found that Thomson’s plum pudding model strikingly disagreed with the results. He then offered a competing model featuring a dense, positive core surrounded by quickly moving electrons.12 This paper intends to outline Rutherford’s experiment with a modern apparatus and prove the necessity of a dense and positive atomic nucleus. II. EXPERIMENT Our experimental apparatus consists of a scattering chamber, americium source, gold foil, and detector with discriminator and counting machinery. The scattering chamber was evacuated to 650 torr, and within which was placed the americium source, gold foil, and detector as shown in figure 1. The detector is free to move an angle θ from the direct path of the α-particles. We measured the number of counts per time as a function of the detection angle. See section IV for resulting data and analysis. Rutherford Scattering III. 2 equation (5) to yield THEORY In this paper we will treat the Rutherford scattering problem with a modern approach and take the classical limit.13 The problem to solve is the scattering of an α-particle off a much more massive gold nucleus. The interaction involved is Coulombic repulsion (1) where Z1 and Z2 are the atomic numbers of the colliding atoms, and e is the electronic charge. We will generalize this potential with an exponential factor that decreases with radius to account for interactions with the electron cloud on gold e−m0 r r µ 2π~2 Z 2µZ1 Z2 e2 . (m20 + q 2 ) ~2 We may then return q to our initial variables by q 2 = (ki − kf ) . 0 0 (8) and since ki = kf by momentum conservation, q 2 = 2k 2 (1 − cos θ) θ 2 2 . = 4k sin 2 (2) d3 r0 e−ikf ·r V (r 0 ) eiki ·r , 2 = ki2 + kf2 + 2ki · kf , From the Born approximation, the scattering amplitude f (θ, ϕ) is given by f (θ, ϕ) = − =− (7) Z1 Z2 e2 , V (r) = r V (r) = Z1 Z2 e2 Z Z Z 0 0 µZ1 Z2 e2 ∞ 0 2π 0 π 0 dθ sin θ0 r0 e−m0 r eiqr cos θ0 dϕ dr 2π~2 0 0 0 Z µZ1 Z2 e2 i ∞ −m0 r0 iqr0 −iqr 0 e e − e = ~2 q 0 f (θ, ϕ) = − (9) Thus we may substitute equation (9) into our scattering amplitude (7) to yield (3) f (θ, ϕ) = − 2µZ1 Z2 e2 2 ~2 m0 + 4k 2 sin2 θ 2 . (10) where ki is the direction of the incident matter wave, kf is the direction of the resulting matter wave, and µ the mass of the particle. The scattering amplitude f (θ, ϕ) relates to the differential cross section by By equation (4), we then have the differential cross section dσ 2 = |f (θ, ϕ)| , dΩ dσ 4µ2 Z12 Z22 e4 h i. = 2 dΩ ~4 m20 + 4k 2 sin2 (θ/2) (4) the derivative of the total cross section with respect to solid angle. To obtain our scattering amplitude, we will then substitute our potential in equation (2) into equation (3) µZ1 Z2 e2 2π~2 Z f (θ, ϕ) = − =− µZ1 Z2 e2 2π~2 Z The α-particles will have a kinetic energy on the order of 5 MeV. Then ~2 k 2 ' 5 MeV, (12) k 2 ' 7.2 · 1055 (m · kg) −2 . (13) 0 d3 r0 e−ikf ·r 0 d3 r 0 0 e−m0 r iki ·r0 e r0 e−m0 r iq·r0 e , r0 so (5) where we have defined q ≡ ki − kf . The inner product q · r 0 may be evaluated using the angle θ0 between the detector and initial path Then we may safely assume 4k 2 sin2 (θ/2) m0 , 0 (11) 0 0 q · r = qr cos θ , (14) (6) which we may then substitute into our integral in since the Coulombic interaction contribution from the electron cloud is of negligible order in comparison. Then Rutherford Scattering 3 we may reduce equation (11) to dσ µ2 Z 2 Z 2 e4 . = 4 4 1 42 dΩ 4k ~ sin (θ/2) (15) Finally, the energy of a wave is given by E = ~2 k 2 , so we receive the differential cross section for our system dσ = dΩ µZ1 Z2 e2 4E sin2 (θ/2) 2 (16) which gives the number of α-particles deflected into an angle θ N (θ) = 1 4π0 2 Z1 Z2 e2 2µv 2 2 1 , sin4 (θ/2) (17) FIG. 2. Counts per second as a function of detector angle. in SI units and v is the velocity of an α-particle. After substitution of fundamental constants and Z1 = 79 and Z2 = 4, N (θ) ' v2 7.4217 . sin4 (θ/2) (18) The energy of an α-particle according to the lab manual is E ' 5.48 MeV, so mv 2 /2 = 5.48 MeV means N (θ) ' 2.828 · 1014 . sin4 (θ/2) (19) FIG. 3. Linear regression on counts per second versus 1/ sin4 (angle/2). IV. ANALYSIS We collected data over the period of four weeks ranging from angles of -60 to +65 degrees. Using the fact that radioactive decay follows a Poisson distribution, the standard deviation grows with the square root of the average number of counts. These standard deviations are shown on figure 2. Observe the strong central peak that quickly decays on either side of 0 angle. To check if our data agree with the theoretically predicted sin4 (θ/2) trend, we performed a linear regression on counts per second versus sin4 (angle/2) as shown in figure 3. This regression yields R2 = 0.885, meaning that the data roughly agree with the prediction. only know our angle to a precision of 2.5 degrees. The detector also works as a photon detector and even a vibrations detector at times. For this reason, stray photons (or vibrations) may have contributed to the α-particle count. Although we maintained the scattering chamber at 650 torr throughout the experiment, this vacuum is not sufficient for entirely pristine results – stray air molecules affect the trajectories of the α-particles. The gold foil used in this experiment is more than a single atom thickness, and there were very likely many scattering events for each α particle although our analysis describes only one. V. VI. UNCERTAINTY For this experiment, we used a dial to alter the detector angle. The dial has tick marks every 5 degrees, so we can CONCLUSION We derived the expected differential cross section for the Coulomb potential using the Born approximation. Rutherford Scattering This prediction was then tested against four weeks of measurements with angles ranging from -60 to +65 degrees in small steps. From these measurements, we found a 1/sin4 (θ/2) relationship between α-particle counts per second and measured angle which agrees with our theoretically predicted trend. From this experiment we see that the α-particles have encountered a very dense, positive potential while passing through the gold foil. This potential is not expected under the Thomson plum pudding model. We then require the atom to have a very dense positive nucleus with electrons in orbit. VII. 1 J. REFERENCES Dalton, A New System of Chemical Philosophy (William Dalton and Sons, Ltd., 1808) Chap. 2, p. 142. 2 L. Boltzmann, Vorlesungen der Gastheorie (Verlag von Johann Ambrosia Barth, 1896). 3 J. B. Perrin, Les Principes (Imprimerie Gauthier-Villars, year=). 4 A. Einstein, “On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat,” Annalen der Physik , 649–560 (1905). 4 5 Gottfried and Yau, Quantum Mechanics: Fundamentals, 2nd ed. (Springer, 2003). 6 C. Herndon and N. Lucas, “Advanced lab: Wave particle duality,” Advanced Lab (2015). 7 J. J. Thomson, “On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure,” Philosophical Magazine Series , 237–265 (1904). 8 E. Rutherford, “Uranium radiation and the electrical conduction produced by it,” Philosophical Magazine (1899). 9 Rutherford and Royds, “Spectrum of the radium emanation,” Nature 78, 220–221 (1908). 10 M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966). 11 Gamow and Cleveland, Physics Foundations and Frontiers (Prentice-Hall, inc., 1960) pp. 365–369. 12 E. Rutherford, “The scattering of α and β-particles by matter and the structure of the atom,” Philosophical Magazine 21, 669– 688 (1911). 13 J. Townsend, A Modern Approach to Quantum Mechanics, 2nd ed. (University Science Books, 2012).