The Rutherford scattering experiments were a landmark series of experiments by which scientists learned that every atom has a nucleus where all of its positive charge and most of its mass is concentrated. They deduced this after measuring how an alpha particle beam is scattered when it strikes a thin metal foil. The experiments were performed between 1906 and 1913 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester. The physical phenomenon was explained by Rutherford in a classic 1911 paper that eventually led to the widespread use of scattering in particle physics to study subatomic matter. Rutherford scattering or Coulomb scattering is the elastic scattering of charged particles by the Coulomb interaction. The paper also initiated the development of the planetary Rutherford model of the atom and eventually the Bohr model.
Summary
Thomson's model of the atom
The prevailing model of atomic structure before Rutherford's experiments was devised by J. J. Thomson. Thomson had discovered the electron through his work on cathode rays and proposed that they existed within atoms, and an electric current is electrons hopping from one atom to an adjacent one in a series. There logically had to be a commensurate amount of positive charge to balance the negative charge of the electrons and hold those electrons together. Having no idea what the source of this positive charge was, he tentatively proposed that the positive charge was everywhere in the atom, adopting a spherical shape for simplicity. Thomson imagined that the balance of electrostatic forces would distribute the electrons throughout this sphere in a more or less even manner. Thomson also believed the electrons could move around in this sphere, and in that regard, he likened the substance of the sphere to a liquid. The positive sphere was more of an abstraction than anything material. He did not propose a positively-charged subatomic particle; a counterpart to the electron. Thomson was never able to develop a complete and stable model that could predict any of the other known properties of the atom, such as emission spectra and valencies. The Japanese scientist Hantaro Nagaoka rejected Thomson's model on the grounds that opposing charges cannot penetrate each other. He proposed instead that electrons orbit the positive charge like the rings around Saturn. However, this model was also known to be unstable.
Alpha particles and the Thomson atom An alpha particle is a positively charged particle of matter that is spontaneously emitted from certain radioactive elements. Alpha particles are so tiny as to be invisible, but they can be detected with the use of phosphorescent screens, photographic plates, or electrodes. Rutherford discovered them in 1899. In 1906, by studying how alpha particle beams are deflected by magnetic and electric fields, he deduced that they were essentially helium atoms stripped of two electrons. Thomson and Rutherford knew nothing about the internal structure of alpha particles. At the time, scientists did not know exactly how many electrons a helium atom had (nor atoms of other elements for that matter), so a helium atom stripped of two electrons might still have ten or so left for all they could tell. Thomson's model was consistent with the experimental evidence available at the time. Thomson studied beta particle scattering, which showed small-angle deflections modelled as interactions of the particle with many atoms in succession. Each interaction of the particle with the electrons of the atom and the positive background sphere would lead to a tiny deflection, but many such collisions could add up. The scattering of alpha particles was expected to be similar. Rutherford's team would show that the multiple scattering model was not needed: single scattering from a compact charge at the centre of the atom would account for all of the scattering data.
Rutherford, Geiger, and Marsden
Ernest Rutherford was Langworthy Professor of Physics at the Victoria University of Manchester (now the University of Manchester). He had already received numerous honours for his studies of radiation. He had discovered the existence of alpha rays, beta rays, and gamma rays, and had proved that these were the consequence of the disintegration of atoms. In 1906, he received a visit from the German physicist Hans Geiger, and was so impressed that he asked Geiger to stay and help him with his research. Ernest Marsden was a physics undergraduate student studying under Geiger. In 1908, Rutherford sought to determine the charge and mass of alpha particles independently. To do this, he wanted to count the number of alpha particles and measure their total charge; the ratio would give the charge of a single alpha particle. Alpha particles are too tiny to see, but Rutherford knew about the Townsend discharge, a cascade effect from ionisation leading to a pulse of electric current. On this principle, Rutherford and Geiger designed a simple counting device which consisted of two electrodes in a glass tube containing low pressure gas. (See #1908 experiment.) Every alpha particle that passed through the gas would create a pulse of electrical current that could be detected and counted. It was the forerunner of the Geiger-Müller Counter. The counter that Geiger and Rutherford built proved unreliable because the alpha particles were being too strongly deflected by their collisions with the molecules of air within the detection chamber. The highly variable trajectories of the alpha particles meant that they did not all generate the same number of ions as they passed through the gas, thus producing erratic readings. This puzzled Rutherford because he had thought that alpha particles were too heavy to be deflected so strongly. Rutherford asked Geiger to investigate how far matter could scatter alpha rays. The experiments they designed involved bombarding metal foil with a beam of alpha particles to observe how the foil scattered them in relation to its thickness and material. They used a phosphorescent screen to measure the trajectories of the particles. Each impact of an alpha particle on the screen produced a tiny flash of light. Geiger worked in a darkened lab for hours on end, counting these tiny scintillations using a microscope. For the metal foil, they tested a variety of metals, but favoured gold because they could make the foil very thin, as gold is the most malleable metal. As a source of alpha particles, Rutherford's substance of choice was radium, which is thousands of times more radioactive than uranium.
Scattering theory and the new atomic model
In a 1909 experiment, Geiger and Marsden discovered that the metal foils could scatter some alpha particles in all directions, sometimes more than 90°. This should have been impossible according to Thomson's model. According to Thomson's model, all the alpha particles should have gone straight through. In Thomson's model of the atom, the sphere of positive charge that fills the atom and encapsulates the electrons is permeable; the electrons could move around in it, after all. Therefore, an alpha particle should be able to pass through this sphere if the electrostatic forces within permit it. Thomson himself did not study how an alpha particle might be scattered in such a collision with an atom, but he did study beta particle scattering. He calculated that a beta particle would only experience very small deflection when passing through an atom, and even after passing through many atoms in a row, the total deflection should still be less than 1°. Alpha particles typically have much more momentum than beta particles and therefore should likewise experience only the slightest deflection. The extreme scattering observed forced Rutherford to revise the model of the atom. The issue in Thomson's model was that the charges were too diffuse to produce a sufficiently strong electrostatic force to cause such repulsion. Therefore they had to be more concentrated. In Rutherford's new model, the positive charge does not fill the entire volume of the atom but instead constitutes a tiny nucleus at least 10,000 times smaller than the atom as a whole. All that positive charge concentrated in a much smaller volume produces a much stronger electric field near its surface. The nucleus also carried most of the atom's mass. This meant that it could deflect alpha particles by up to 180° depending on how close they pass. The electrons surround this nucleus, spread throughout the atom's volume. Because their negative charge is diffuse and their combined mass is low, they have a negligible effect on the alpha particle. To verify his model, Rutherford developed a scientific model to predict the intensity of alpha particles at the different angles they scattered coming out of the gold foil, assuming all of the positive charge was concentrated at the centre of the atom. This model was validated in an experiment performed in 1913. His model explained both the beta scattering results of Thomson and the alpha scattering results of Geiger and Marsden.
Legacy There was little reaction to Rutherford's now-famous 1911 paper in the first years. The paper was primarily about alpha particle scattering in an era before particle scattering was a primary tool for physics. The probability techniques he used and confusing collection of observations involved were not immediately compelling.
Nuclear physics
The first impacts were to encourage new focus on scattering experiments. For example the first results from a cloud chamber, by C.T.R. Wilson shows alpha particle scattering and also appeared in 1911. Over time, particle scattering became a major aspect of theoretical and experimental physics; Rutherford's concept of a "cross section" now dominates the descriptions of experimental particle physics. The historian Silvan S. Schweber suggests that Rutherford's approach marked the shift to viewing all interactions and measurements in physics as scattering processes. After the nucleus - a term Rutherford introduced in 1912 - became the accepted model for the core of atoms, Rutherford's analysis of the scattering of alpha particles created a new branch of physics, nuclear physics.
Atomic model
Rutherford's new atom model caused no stir. Rutherford explicitly ignores the electrons, only mentioning Hantaro Nagaoka's Saturnian model of electrons orbiting a tiny "sun", a model that had been previously rejected as mechanically unstable. By ignoring the electrons Rutherford also ignores any potential implications for atomic spectroscopy for chemistry. Rutherford himself did not press the case for his atomic model: his own 1913 book on "Radioactive substances and their radiations" only mentions the atom twice; other books by other authors around this time focus on Thomson's model. The impact of Rutherford's nuclear model came after Niels Bohr arrived as a post-doctoral student in Manchester at Rutherford's invitation. Bohr dropped his work on the Thomson model in favour of Rutherford's nuclear model, developing the Rutherford–Bohr model over the next several years. Eventually Bohr incorporated early ideas of quantum mechanics into the model of the atom, allowing prediction of electronic spectra and concepts of chemistry. Hantaro Nagaoka, who had proposed a Saturnian model of the atom, wrote to Rutherford from Tokyo in 1911: "I have been struck with the simpleness of the apparatus you employ and the brilliant results you obtain." The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier. Rutherford has since been hailed as "the father of nuclear physics". In a lecture delivered on 15 October 1936 at Cambridge University, Rutherford described his shock at the results of the 1909 experiment:
Then I remember two or three days later Geiger coming to me in great excitement and saying, "We have been able to get some of the α-particles coming backwards...". It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realised that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive centre, carrying a charge. Rutherford's claim of surprise makes for a good story but by the time of the Geiger-Marsden experiment, the result confirmed suspicions Rutherford developed from previous experiments.
Experiments
Alpha particle scattering: 1906 and 1908 experiments Rutherford's first steps towards his discovery of the nature of the atom came from his work to understand alpha particles. In 1906, Rutherford noticed that alpha particles passing through sheets of mica were deflected by the sheets by as much as 2 degrees. Rutherford placed a radioactive source in a sealed tube ending with a narrow slits followed by a photographic plate. Half of the slit was covered by a thin layer of mica. A magnetic field around the tube was altered every 10 minutes to reject the effect of beta rays, known to be sensitive to magnetic fields. The tube was evacuated to different amounts and a series of images recorded. At the lowest pressure the image of the open slit was clear, while images of the mica covered slit or the open slit at higher pressures were fuzzy. Rutherford explained these results as alpha-particle scattering in a paper published in 1906. He already understood the implications of the observation for models of atoms: "such a result brings out clearly the fact that the atoms of matter must be the seat of very intense electrical forces".
A 1908 paper by Geiger, On the Scattering of α-Particles by Matter, describes the following experiment. He constructed a long glass tube, nearly two metres long. At one end of the tube was a quantity of "radium emanation" (R) as a source of alpha particles. The opposite end of the tube was covered with a phosphorescent screen (Z). In the middle of the tube was a 0.9 mm-wide slit. The alpha particles from R passed through the slit and created a glowing patch of light on the screen. A microscope (M) was used to count the scintillations on the screen and measure their spread. Geiger pumped all the air out of the tube so that the alpha particles would be unobstructed, and they left a neat and tight image on the screen that corresponded to the shape of the slit. Geiger then allowed some air into the tube, and the glowing patch became more diffuse. Geiger then pumped out the air and placed one or two gold foils over the slit at AA. This too caused the patch of light on the screen to become more spread out, with the larger spread for two layers. This experiment demonstrated that both air and solid matter could markedly scatter alpha particles.
Alpha particle reflection: the 1909 experiment
The results of the initial alpha particle scattering experiments were confusing. The angular spread of the particle on the screen varied greatly with the shape of the apparatus and its internal pressure. Rutherford suggested that Ernest Marsden, a physics undergraduate student studying under Geiger, should look for diffusely reflected or back-scattered alpha particles, even though these were not expected. Marsden's first crude reflector got results, so Geiger helped him create a more sophisticated apparatus. They were able to demonstrate that 1 in 8000 alpha particle collisions were diffuse reflections. Although this fraction was small, it was much larger than the Thomson model of the atom could explain.
These results were published in a 1909 paper, On a Diffuse Reflection of the α-Particles, where Geiger and Marsden described the experiment by which they proved that alpha particles can indeed be scattered by more than 90°. In their experiment, they prepared a small conical glass tube (AB) containing "radium emanation" (radon), "radium A" (actual radium), and "radium C" (bismuth-214); its open end was sealed with mica. This was their alpha particle emitter. They then set up a lead plate (P), behind which they placed a fluorescent screen (S). The tube was held on the opposite side of plate, such that the alpha particles it emitted could not directly strike the screen. They noticed a few scintillations on the screen because some alpha particles got around the plate by bouncing off air molecules. They then placed a metal foil (R) to the side of the lead plate. They tested with lead, gold, tin, aluminium, copper, silver, iron, and platinum. They pointed the tube at the foil to see if the alpha particles would bounce off it and strike the screen on the other side of the plate, and observed an increase in the number of scintillations on the screen. Counting the scintillations, they observed that metals with higher atomic mass, such as gold, reflected more alpha particles than lighter ones such as aluminium. Geiger and Marsden then wanted to estimate the total number of alpha particles that were reflected. The previous setup was unsuitable for doing this because the tube contained several radioactive substances (radium plus its decay products) and thus the alpha particles emitted had varying ranges, and because it was difficult for them to ascertain at what rate the tube was emitting alpha particles. This time, they placed a small quantity of radium C (bismuth-214) on the lead plate, which bounced off a platinum reflector (R) and onto the screen. They concluded that approximately 1 in 8,000 of the alpha particles that struck the reflector bounced onto the screen. By measuring the reflection from thin foils they showed that the effect was due to a volume and not a surface effect. When contrasted with the vast number of alpha particles that pass unhindered through a metal foil, this small number of large angle reflections was a strange result that meant very large forces were involved.
Dependence on foil material and thickness: the 1910 experiment
A 1910 paper by Geiger, The Scattering of the α-Particles by Matter, describes an experiment to measure how the most probable angle through which an alpha particle is deflected varies with the material it passes through, the thickness of the material, and the velocity of the alpha particles. He constructed an airtight glass tube from which the air was pumped out. At one end was a bulb (B) containing "radium emanation" (radon-222). By means of mercury, the radon in B was pumped up the narrow glass pipe whose end at A was plugged with mica. At the other end of the tube was a fluorescent zinc sulfide screen (S). The microscope which he used to count the scintillations on the screen was affixed to a vertical millimetre scale with a vernier, which allowed Geiger to precisely measure where the flashes of light appeared on the screen and thus calculate the particles' angles of deflection. The alpha particles emitted from A were narrowed to a beam by a small circular hole at D. Geiger placed a metal foil in the path of the rays at D and E to observe how the zone of flashes changed. He tested gold, tin, silver, copper, and aluminium. He could also vary the velocity of the alpha particles by placing extra sheets of mica or aluminium at A. From the measurements he took, Geiger came to the following conclusions:
the most probable angle of deflection increases with the thickness of the material the most probable angle of deflection is proportional to the atomic mass of the substance the most probable angle of deflection decreases with the velocity of the alpha particles
Rutherford's Structure of the Atom paper (1911)
Considering the results of these experiments, Rutherford published a landmark paper in 1911 titled "The Scattering of α and β Particles by Matter and the Structure of the Atom" wherein he showed that single scattering from a very small and intense electric charge predicts primarily small-angle scattering with small but measurable amounts of backscattering. For the purpose of his mathematical calculations he assumed this central charge was positive, but he admitted he could not prove this and that he had to wait for other experiments to develop his theory. Rutherford developed a mathematical equation that modelled how the foil should scatter the alpha particles if all the positive charge and most of the atomic mass was concentrated in a point at the centre of an atom. From the scattering data, Rutherford estimated the central charge qn to be about +100 units. Rutherford's paper does not discuss any electron arrangement beyond discussions on the scattering from Thomson's plum pudding model and Nagaoka's Saturnian model. He shows that the scattering results predicted by Thomson's model are also explained by single scattering, but that Thomson's model does not explain large angle scattering. He says that Nagaoka's model, having a compact charge, would agree with the scattering data. But the Saturnian model had previously been rejected on other grounds. The so-called Rutherford model of the atom with orbiting electrons was not proposed by Rutherford in the 1911 paper.
Confirming the scattering theory: the 1913 experiment In a 1913 paper, The Laws of Deflexion of α Particles through Large Angles, Geiger and Marsden describe a series of experiments by which they sought to experimentally verify Rutherford's equation. Rutherford's equation predicted that the number of scintillations per minute s that will be observed at a given angle Φ should be proportional to:
cosec4Φ/2 thickness of foil t magnitude of the square of central charge Qn 1/(mv2)2 Their 1913 paper describes four experiments by which they proved each of these four relationships.
To test how the scattering varied with the angle of deflection (i.e. if s ∝ csc4Φ/2). Geiger and Marsden built an apparatus that consisted of a hollow metal cylinder mounted on a turntable. Inside the cylinder was a metal foil (F) and a radiation source containing radon (R), mounted on a detached column (T) which allowed the cylinder to rotate independently. The column was also a tube by which air was pumped out of the cylinder. A microscope (M) with its objective lens covered by a fluorescent zinc sulfide screen (S) penetrated the wall of the cylinder and pointed at the metal foil. They tested with silver and gold foils. By turning the table, the microscope could be moved a full circle around the foil, allowing Geiger to observe and count alpha particles deflected by up to 150°. Correcting for experimental error, Geiger and Marsden found that the number of alpha particles that are deflected by a given angle Φ is indeed proportional to csc4Φ/2.
Geiger and Marsden then tested how the scattering varied with the thickness of the foil (i.e. if s ∝ t). They constructed a disc (S) with six holes drilled in it. The holes were covered with metal foil (F) of varying thickness, or none for control. This disc was then sealed in a brass ring (A) between two glass plates (B and C). The disc could be rotated by means of a rod (P) to bring each window in front of the alpha particle source (R). On the rear glass pane was a zinc sulfide screen (Z). Geiger and Marsden found that the number of scintillations that appeared on the screen was indeed proportional to the thickness, as long as the thickness was small. Geiger and Marsden reused the apparatus to measure how the scattering pattern varied with the square of the nuclear charge (i.e. if s ∝ Qn2). Geiger and Marsden did not know what the positive charge of the nucleus of their metals were (they had only just discovered the nucleus existed at all), but they assumed it was proportional to the atomic weight, so they tested whether the scattering was proportional to the atomic weight squared. Geiger and Marsden covered the holes of the disc with foils of gold, tin, silver, copper, and aluminium. They measured each foil's stopping power by equating it to an equivalent thickness of air. They counted the number of scintillations per minute that each foil produced on the screen. They divided the number of scintillations per minute by the respective foil's air equivalent, then divided again by the square root of the atomic weight (Geiger and Marsden knew that for foils of equal stopping power, the number of atoms per unit area is proportional to the square root of the atomic weight). Thus, for each metal, Geiger and Marsden obtained the number of scintillations that a fixed number of atoms produce. For each metal, they then divided this number by the square of the atomic weight, and found that the ratios were about the same. Thus they proved that s ∝ Qn2. Finally, Geiger and Marsden tested how the scattering varied with the velocity of the alpha particles (i.e. if s ∝ 1/v4). Using the same apparatus, they slowed the alpha particles by placing extra sheets of mica in front of the alpha particle source. They found that, within the range of experimental error, the number of scintillations was indeed proportional to 1/v4.
Positive charge on nucleus: 1913 In his 1911 paper (see above), Rutherford assumed that the central charge of the atom was positive, but a negative charge would have fitted his scattering model just as well. In a 1913 paper, Rutherford declared that the "nucleus" (as he now called it) was indeed positively charged, based on the result of experiments exploring the scattering of alpha particles in various gases. In 1917, Rutherford and his assistant William Kay began exploring the passage of alpha particles through gases such as hydrogen and nitrogen. In this experiment, they shot a beam of alpha particles through hydrogen, and they carefully placed their detector—a zinc sulfide screen—just beyond the range of the alpha particles, which were absorbed by the gas. They nonetheless picked up charged particles of some sort causing scintillations on the screen. Rutherford interpreted this as alpha particles knocking the hydrogen nuclei forwards in the direction of the beam, not backwards.
Rutherford's scattering model Rutherford begins his 1911 paper with a discussion of Thomson's results on scattering of beta particles, a form of radioactivity that results in high velocity electrons. Thomson's model had electrons circulating inside of a sphere of positive charge. Rutherford highlights the need for compound or multiple scattering events: the deflections predicted for each collision are much less than one degree. He then proposes a model which will produce large deflections on a single encounter: place all of the positive charge at the centre of the sphere and ignore the electron scattering as insignificant. The concentrated charge will explain why most alpha particles do not scatter to any measurable degree – they fly past too far from the charge – and yet particles that do pass very close to the centre scatter through large angles.
Maximum nuclear size estimate Rutherford begins his analysis by considering a head-on collision between the alpha particle and atom. This will establish the minimum distance between them, a value which will be used throughout his calculations. Assuming there are no external forces and that initially the alpha particles are far from the nucleus, the inverse-square law between the charges on the alpha particle and nucleus gives the potential energy gained by the particle as it approaches the nucleus. For head-on collisions between alpha particles and the nucleus, all the kinetic energy of the alpha particle is turned into potential energy and the particle stops and turns back.
Where the particle stops at a distance r min {\displaystyle r_{\text{min}}} from the centre, the potential energy matches the original kinetic energy:
1 2 m v 2 = k q a q g r min {\displaystyle {\frac {1}{2}}mv^{2}=k{\frac {q_{\text{a}}q_{\text{g}}}{r_{\text{min}}}}}
where
k = 1 4 π ε 0 {\displaystyle k={\frac {1}{4\pi \varepsilon _{0}}}}
Rearranging:
r min = k 2 q a q g m v 2 {\displaystyle r_{\text{min}}=k{\frac {2q_{\text{a}}q_{\text{g}}}{mv^{2}}}}
For an alpha particle:
m (mass) = 6.64424×10−27 kg = 3.7273×109 eV/c2 qa (for the alpha particle) = 2 × 1.6×10−19 C = 3.2×10−19 C qg (for gold) = 79 × 1.6×10−19 C = 1.27×10−17 C v (initial velocity) = 2×107 m/s (for this example) The distance from the alpha particle to the centre of the nucleus (rmin) at this point is an upper limit for the nuclear radius. Substituting these in gives the value of about 2.7×10−14 m, or 27 fm. (The true radius is about 7.3 fm.) The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear centre, as noted, when the radius of the nucleus of a gold atom is 7.3 fm.
Rutherford's 1911 paper started with a slightly different formula suitable for head-on collision with a sphere of positive charge:
1 2 m v 2 = N e E ⋅ ( 1 b − 3 2 R + b 2 2 R 3 ) {\displaystyle {\frac {1}{2}}mv^{2}=NeE\cdot \left({\frac {1}{b}}-{\frac {3}{2R}}+{\frac {b^{2}}{2R^{3}}}\right)}
In Rutherford's notation, e is the elementary charge, N is the charge number of the nucleus (now also known as the atomic number), and E is the charge of an alpha particle. The convention in Rutherford's time was to measure charge in electrostatic units, distance in centimeters, force in dynes, and energy in ergs. The modern convention is to measure charge in coulombs, distance in meters, force in newtons, and energy in joules. Using coulombs requires using the Coulomb constant (k) in the equation. Rutherford used b as the turning point distance (called rmin above) and R is the radius of the atom. The first term is the Coulomb repulsion used above. This form assumes the alpha particle could penetrate the positive charge. At the time of Rutherford's paper, Thomson's plum pudding model proposed a positive charge with the radius of an atom, thousands of times larger than the rmin found above. Figure 1 shows how concentrated this potential is compared to the size of the atom. Many of Rutherford's results are expressed in terms of this turning point distance rmin, simplifying the results and limiting the need for units to this calculation of turning point.
Single scattering by a heavy nucleus From his results for a head on collision, Rutherford knows that alpha particle scattering occurs close to the centre of an atom, at a radius 10,000 times smaller than the atom. The electrons have negligible effect. He begins by assuming no energy loss in the collision, that is he ignores the recoil of the target atom. He will revisit each of these issues later in his paper. Under these conditions, the alpha particle and atom interact through a central force, a physical problem studied first by Isaac Newton. A central force only acts along a line between the particles and when the force varies with the inverse square, like Coulomb force in this case, a detailed theory was developed under the name of the Kepler problem. The well-known solutions to the Kepler problem are called orbits and unbound orbits are hyperbolas. Thus Rutherford proposed that the alpha particle will take a hyperbolic trajectory in the repulsive force near the centre of the atom as shown in Figure 2.
To apply the hyperbolic trajectory solutions to the alpha particle problem, Rutherford expresses the parameters of the hyperbola in terms of the scattering geometry and energies. He starts with conservation of angular momentum. When the particle of mass m {\displaystyle m} and initial velocity v 0 {\displaystyle v_{0}} is far from the atom, its angular momentum around the centre of the atom will be m b v 0 {\displaystyle mbv_{0}} where b {\displaystyle b} is the impact parameter, which is the lateral distance between the alpha particle's path and the atom. At the point of closest approach, labeled A in Figure 2, the angular momentum will be m r A v A {\displaystyle mr_{\text{A}}v_{\text{A}}} . Therefore
m b v 0 = m r A v A {\displaystyle mbv_{0}=mr_{\text{A}}v_{\text{A}}}
v A = b v 0 r A {\displaystyle v_{\text{A}}={\frac {bv_{0}}{r_{\text{A}}}}}
Rutherford also applies the law of conservation of energy between the same two points:
1 2 m v 0 2 = 1 2 m v A 2 + k q a q g r A {\displaystyle {\tfrac {1}{2}}mv_{0}^{2}={\tfrac {1}{2}}mv_{\text{A}}^{2}+{\frac {kq_{\text{a}}q_{\text{g}}}{r_{\text{A}}}}}
The left hand side and the first term on the right hand side are the kinetic energies of the particle at the two points; the last term is the potential energy due to the Coulomb force between the alpha particle and atom at the point of closest approach (A). qa is the charge of the alpha particle, qg is the charge of the nucleus, and k is the Coulomb constant. The energy equation can then be rearranged thus:
v A 2 = v 0 2 ( 1 − k q a q g 1 2 m v 0 2 r A ) {\displaystyle v_{\text{A}}^{2}=v_{0}^{2}\left(1-{\frac {kq_{\text{a}}q_{\text{g}}}{{\tfrac {1}{2}}mv_{0}^{2}r_{\text{A}}}}\right)}
For convenience, the non-geometric physical variables in this equation can be contained in a variable r min {\displaystyle r_{\text{min}}} , which is the point of closest approach in a head-on collision scenario which was explored in a previous section of this article:
r min = k q a q g 1 2 m v 0 2 {\displaystyle r_{\text{min}}={\frac {kq_{\text{a}}q_{\text{g}}}{{\tfrac {1}{2}}mv_{0}^{2}}}}
This allows Rutherford simplify the energy equation to: