Rutherford scattering experiments

thumb|upright=1.3|A replica of an apparatus used by Geiger and Marsden to measure alpha particle scattering in a 1913 experiment

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.

Rutherford scattering is now exploited by the materials science community in an analytical technique called Rutherford backscattering.

Summary

Thomson's model of the atom

thumb|The "[[plum pudding model" of an atom with seven electrons, as imagined by J. J. Thomson in 1905]]

The prevailing model of atomic structure before Rutherford's experiments was devised by J. J. Thomson. 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. 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. (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

upright=1.5|thumb|Left: Had Thomson's model been correct, all the alpha particles should have passed through the foil with minimal scattering. Right: What Geiger and Marsden observed was that a small fraction of the alpha particles experienced strong deflection.

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. 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 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

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. The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier.

In a lecture delivered on 15 October 1936 at Cambridge University, Rutherford described his shock at the results of the 1909 experiment:

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. 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 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".

upright=1.5|thumb|This apparatus was described in a 1908 paper by Hans Geiger. It could only measure deflections of a few degrees.

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. 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.

{r_\text{min</math>

where

<math display="block">k = \frac{1}{4\pi \varepsilon_0}</math>

Rearranging:

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. Therefore:

As can be deduced from Figure 2, the focal distance SO is

and therefore

With these formulas for SO and OA, the distance <math>r_\text{A}</math> can be written in terms of <math>\Phi</math> and simplified using a trigonometric identity known as a half-angle formula: on classical mechanics.

Intensity vs angle

thumb|Geometry of differential scattering cross section

To compare to experiments the relationship between impact parameter and scattering angle needs to be converted to probability versus angle. The scattering cross section gives the relative intensity by angles:

At the end of his development of the cross section formula, Rutherford emphasises that the results apply to single scattering and thus require measurements with thin foils. For thin foils the degree of scattering is proportional to the foil thickness in agreement with Geiger's measurements. which predicted that in the plum pudding model, a beta particle could be scattered by a significant angle after a series of atomic collisions. Rutherford's model produced stronger scattering by concentrating the positive charge of the atom at a central point rather than spread it over the volume of the atom. Then a collision with just one atom could produce a larger effect on a beta particle than Thomson's model. Rutherford completed his analysis including the effects of density and foil thickness, then concluded that thin foils are governed by single collision scattering, not multiple collision scattering. In the lab frame, denoted by a subscript L, the scattering angle for a general central potential is

<math display="block">\tan \Theta_\text{L} = \frac{\sin\Theta}{\cos\Theta + \frac{m_1}{m_2</math>

For a heavy particle like gold used by Rutherford, the factor <math>\tfrac{m_1}{m_2} = \tfrac{4}{197} \approx 0.02</math> can be neglected at almost all angles. Then the lab and relative angles are the same, <math>\Theta_\text{L} \approx \Theta</math>.

The change in scattering angle alters the formula for differential cross section needed for comparison to experiment. In general the calculation is complex. For the case of alpha-particle scattering from gold atoms, this effect on the cross section is quite small. showing the limits of the 1911 formula even with corrections for reduced mass. Similar issues with smaller deviations for helium, magnesium, aluminium led to the conclusion that the alpha particle was penetrating the nucleus in these cases. This allowed the first estimates of the size of atomic nuclei. At the same time, it will adapt these equations to alpha particle scattering on the assumption that alpha particles are point charges much like beta particles. These equations will then be used to show that Thomson's model was inconsistent with the experimental results of Geiger and Marsden.

Partial deflection by the positive sphere

Consider an alpha particle passing by a sphere of pure positive charge (no electrons) with a radius R. The sphere is so much heavier than the alpha particle that we do not account for recoil. Its position is fixed. The alpha particle passes just close enough to graze the edge of the sphere, which is where the electric field of the sphere is strongest.

center|thumb|upright=2|Figure 4

An earlier section of this article presented an equation which models how an incoming charged particle is deflected by another charged particle at a fixed position (ie infinite mass).

<math display="block">\theta = 2 \arctan {\frac{k q_1 q_2}{m v^2 b</math>

This equation can be used to calculate the deflection angle in the special case in Figure 4 by setting the impact parameter b to the same value as the radius of the sphere R. So long as the alpha particle does not penetrate the sphere, there is no difference between a sphere of charge and a point charge, a mathematical result known as the Shell theorem.

qg = positive charge of the gold atom = =
qa = charge of the alpha particle = =
R = radius of the gold atom =
v = speed of the alpha particle =
m = mass of the alpha particle =
k = Coulomb constant =

<math display="block">\theta_2 = 2 \arctan {\frac{k q_\text{a} q_\text{g{m v^2 R \approx 0.02 \text{ degrees}</math>

This shows that the largest possible deflection will be very small, to the point that the path of the alpha particle passing through the positive sphere of a gold atom is almost a straight line. Therefore in computing the average deflection, which will be smaller still, we will treat the particle's path through the sphere as a chord of length L.

center|thumb|upright=2|Figure 5

Inside a sphere of uniformly distributed positive charge, the force exerted on the alpha particle at any point along its path through the sphere is

<math display="block">e^{-(90 / 0.8)^2} \approx e^{-12656} \approx 10^{-5946}</math>

While in Thomson's plum pudding model it is mathematically possible that an alpha particle could be deflected by more than 90° after 10,000 collisions, the probability of such an event is so low as to be undetectable. Geiger and Marsden should not have detected any alpha particles coming back in the experiment they performed in 1909, and yet they did.

Notes on historical measurements

Rutherford assumed that the radius of atoms in general to be on the order of 10−10 m and the positive charge of a gold atom to be about 100 times that of hydrogen (). From an experiment in 1906, Rutherford measured alpha particles to have a charge of and an atomic weight of 4, and alpha particles emitted by radon to have velocity of . Rutherford deduced that alpha particles are essentially helium atoms stripped of two electrons, but at the time scientists only had a rough idea of how many electrons atoms have and so the alpha particle was thought to have up to 10 electrons left. In 1909 Robert A. Millikan provided a more accurate measurement of , only 0.6% off the current accepted measurement. Jean Perrin in 1909 measured the mass of the hydrogen atom to be , and if an alpha particle is four times as heavy as that, it would have an absolute mass of .

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 in certain equations. In this article, Rutherford and Thomson's equations have been rewritten to fit modern notation conventions.

Rutherford scattering experiments

Summary

Thomson's model of the atom

Scattering theory and the new atomic model

Intensity vs angle

Partial deflection by the positive sphere

Notes on historical measurements

See also

References

Bibliography

External links