Rutherford scattering of α-particles from gold foils Tony Hyun Kim∗ MIT Department of Physics (Dated: October 16, 2008) We perform α-particle scattering off of gold nuclei. The angular dependence of the scattering rate is measured and is shown to be in good agreement (χ2=dof = 0:93) with the Rutherford differential cross section, thus validating the concept of the atomic nucleus. The dependence of the rate (at fixed angle) on the incident energy is also considered, providing further verification of the Rutherford formula. (χ2=dof = 1:45) 1. INTRODUCTION in the Coulombic repulsion of the incident α-particle from a fixed, positive nucleus. (The electrons are too light to By the beginning of the 20th century, the physics com- deflect α-particles, while the nucleus is too heavy to be munity had undertaken an investigation of atomic struc- deflected by them.) We have then reduced the scatter- ture. In particular, the discovery of the electron by J.J. ing problem to the well-understood Kepler problem of Thomson, the recognition of its subatomic mass, and its motion in a 1=r potential. detection from various substances all led to the general Under the assumption that the target is uniformly illu- understanding that this negatively-charged particle was minated with projectiles, it is possible to derive from the a component of atoms. These advances initiated spec- Keplerian orbits the scattering cross-section. The cross- ulations on the distribution of the compensating posi- section is proportional to the expected scattering rate at tive charge in the overall neutral atom. Of these, the an angle θ from the incident direction. As shown in [1], most prominent was the diffuse charge model promoted the Rutherford cross-section is: by Thomson, in which the positive charge is uniformly 2 distributed over the entire volume of the atom. dσ k 1 = · 4 (1) In this report, we investigate the angular distribution dΩ 4E sin (θ=2) of α-particle scattering off of gold atoms, originally per- 0 2 formed in 1909 by Geiger and Marsden under the direc- where k = ZZ e is the appropriate Coulomb factor be- 4π0 tion of Rutherford. It is shown that the angular scatter- tween the α-particle and the target nucleus; and E is the ing rates are consistent with an atomic model in which initial kinetic energy of the incident α-particles. the heavy positive charge is assumed to be concentrated In contrast to other atomic models that assume a dis- at a point, called the nucleus, relative to the volume tributed positive charge within the atom, the Ruther- of the atom. Our result strictly contradicts the diffuse ford model is remarkable in that it predicts measurably charge model, and serves as one of the experimental foun- finite scattering rates even at large deflection angles. dations for the subsequent quantum atomic theory. For instance, the \cross-section" corresponding to the As further verification of the Rutherford scattering Thomson model is a rapidly decaying Gaussian, which model, we have also probed the dependence of the scat- predicts essentially no scattering beyond a few degrees tering rate on the incident α-particle energy (at a fixed from gold targets.[2] Hence, the existence of large-angle angle). Here, we again observe plausible agreement with \back-scattering" is crucial in validating the point charge the Rutherford formula. model. 2. THEORETICAL BACKGROUND 2.2. Energy loss of α-particles in matter 2.1. Scattering of α-particles from the nucleus Under the assumption of a heavy, fixed nucleus, the α- particle does no work on the target atom. On the other Earlier studies of the electron have shown its mass to hand, electrons are easily deflected by the α-particles and be negligible relative to the mass of any atom. It was also therefore absorbs energy from the incident beam.[3] For known that α-particles were helium ions, which are also instance, α-particles can penetrate only about 4cm of air much less massive than the typical target atoms used in at STP due to this effect.[2] this experiment. Under the nuclear hypothesis, the essen- With the aid of material property tables such as the tial physics of the scattering experiment is then contained NIST database[4], it is then possible to perform metrol- ogy (calculation of foil thicknesses, etc.) using this phe- nomenon. In our investigation, we have used this effect primarily to generate beams of lesser energy, in order ∗Electronic address: [email protected]; URL: http://web.mit.edu/ to probe the energy dependence of the Rutherford cross kimt/www/ section (Eq.1). 2 3. EXPERIMENTAL SETUP dition to the channel of the dominant 5:486MeV parti- cles. This linear interpretation was assessed by energy The main apparatus, consisting of the 241Am α- loss measurements through foils of known composition particle source, the target gold foils, and the detector and thickness. The result of this calibration is shown in were arranged inside a vacuum chamber as shown in Fig. TableI. 1. Most of our measurements were taken with the vac- uum pressure less than 80 microns, where the energy loss TABLE I: Thicknesses of various foils calculated by assuming suffered by the α-particles through air is neglegible. a linear energy/MCA-channel relationship. Target foil Calculated (µm) Listed (µm) a Au 11:58 ± 0:47 13 ± 1 Ti 5:44 ± 1:01 7:6 Ti (Reducer #1) 7:65 ± 1:02 7:6 Ti (Reducer #2) 13:49 ± 0:85 15:8 Fe 8:03 ± 0:74 6:35 Al #1 20:16 ± 1:86 12:7 Al #2 20:06 ± 1:53 12:7 aWe have less confidence in the correctness of the listed thickness for foils other than gold. While the gold thickness was provided by the Junior lab staff, the others were listed on the targets, which looked old and possibly damaged. In addition, there were no un- certainties given on the thicknesses of these other foils. FIG. 1: The α-particle source, the target and the detector 4. ANALYSIS AND RESULTS inside the vacuum chamber. The detector was fixed in the chamber, while the target holder and the howitzer could be rotated independently. Diagram taken from [2] 4.1. Identification of valid scattering events The 241Am source provided a 5:486MeV (86%), During the experiment, it was clear that the detector 5:443MeV (12:7%) α-particle beam. On the side of the was susceptible to environmental factors such as the room barrel, there were slots in which we could place thin foils lighting. Hence, it was important to have a means of in order to further reduce the energy of the output beam. distinguishing the counts actually generated by the scat- The entire howitzer apparatus could be rotated with re- tered α-particles. Fortunately, at low scattering angles spect to the detector. (θ < 20◦), the count rates were high, and it was possi- While the vacuum chamber was necessary in order to ble to clearly distinguish the peak due to the 5:486MeV conduct the experiment, one severe drawback of the setup beam as shown in Fig.2. We were able to characterize was that the angular position of the howitzer φ was very the peak with a Gaussian fit. The final count was then difficult to assess from outside of the chamber. We esti- obtained by summing all counts within two standard de- mate our accuracy in setting φ to no more than ±1◦. viations from the calculated mean. In addition, the experimental sketch in Fig.1 high- However, for larger angles, the expected rate of scat- lights complications not accounted for by Eq.1. Due tering is significantly lower, and it becomes very difficult to the angular spread of the α-particle beam, and the fi- to identify the valid scatterers from the baseline level. nite sizes of the target and detector, the actual scattering We resolved this issue by selecting a fixed range of valid angle θ does not necessarily equal the howitzer angle φ. channels to be applied to every run. We arrived at a Instead, we expect a howitzer configuration φ0 to yield consensus of MCA channels 971 : 1275 after consider- detection counts corresponding to some spread of angles ing many Gaussian fits of data sets displaying obvious θ about φ0. In the following section, we will partially peaks. At every angle, we then systematically collected correct for this finite geometry effect by convolving the the counts over this fixed interval of MCA channels. Rutherford cross-section with a \smearing function" that describes the spread of the α-particles from the source. Finally, the detector output was converted into pulses 4.2. Angular profile of the α-beam whose heights corresponded to the energy of the detected particle. These pulses were then recorded by a PC-based The sketch of the experimental setup (Fig.1) shows MCA. We have assumed that the MCA channels repre- that the scattering angle θ may not match the howitzer sent a linear scale for the energy. The scale was con- position φ exactly. In order to account for this effect, we structed using the zero channel for zero energy, in ad- investigated the angular spread of the α-particle beam by 3 generated from the angular spread measurement (Fig. 3) as the appropriate \smearing function." Then, the expected count rate as a function of the howitzer position is given as a convolution: Z π −4 C(φ) = C0 g(φ − θ) · sin (θ=2) · dθ (2) 0 where C0 is a constant of proportionality, and the sin−4(θ=2) factor originates from the underlying Ruther- ford cross-section.