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Rutherford's 1911 theory paper

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.

Figure 1. Potential energy diagram for Rutherford's atom model illustrating concentration in the nucleus.

At the time of Rutherford's paper, Thomson's plum pudding model proposed a positive charge with the radius of an atom. By concentrating all of the positive charge in the center of the atom, Rutherford creates a very different model. Figure 1 shows how much more concentrated Rutherford's potential is compared to the size of the atom.

Maximum nuclear size estimate

Rutherford begins his mathematical 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.

Schematic view of a head-on collision between an alpha particle and an atom. The radius of the atom is on the order of 10⁻¹⁰ m and the minimum stopping distance is on the order of 10⁻¹⁴ m.

Many of Rutherford's results are expressed in terms of this turning point distance r_min, simplifying the results and limiting the need for units to this calculation of turning point. For the gold atom, Rutherford's turning point was about 2.7×10⁻¹⁴ m, or 27 fm. (The actual 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 actual radius of gold is 7.3 fm.

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.

Figure 2. The geometry of Rutherford's scattering formula, based on a diagram in his 1911 paper. The alpha particle is the green dot and moves along the green path, which is a hyperbola with O as its centre and S as its external focus. The atomic nucleus is located at S. A is the apsis, the point of closest approach. b is the impact parameter, the lateral distance between the alpha particle's initial trajectory and the nucleus.

Each trajectory is characterized by an impact parameter. Large impact parameters mean the alpha particle is always far from the atom center and remain almost in a straight line. Smaller impact parameters result in larger deflection angles.

Hyperbolic trajectories of alpha particles scattering from a Gold nucleus (modern radius shown as gray circle) as described in Rutherford's 1911 paper

Rutherford gives some illustrative values of the deflection angle $\theta$ vs the ratio of impact parameter b to closest approach distance $r_{\text{min}}$, as shown in this table:

Rutherford's angle of deviation table

$b/r_{\text{min}}$	10	5	2	1	0.5	0.25	0.125
$\theta$	5.7°	11.4°	28°	53°	90°	127°	152°

Rutherford's approach to this scattering problem remains a standard treatment in textbooks on classical mechanics.

Intensity vs angle

Geometry of differential scattering cross-section

Since only a few trajectories come close to the centre, only a few alpha particles are strongly deflected. To make this quantitative, Rutherford sums the trajectories heading in each backscattering angle ($d\Omega$), weighted by the number of alpha particles having the corresponding impact parameter, ($db$) as shown in the diagram.

Rutherford scattering cross-section is strongly peaked around zero degrees, and yet has nonzero values out to 180 degrees.

The resulting formula formula predicted the results that Geiger measured in the coming year. The scattering probability into small angles greatly exceeds the probability in to larger angles, reflecting the tiny nucleus surrounded by empty space. However, for rare close encounters, large angle scattering occurs with just a single target. This matches the experimental results of Marsden and Geiger without the need for multiple scattering.

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.

Comparison to JJ Thomson's results

See also: Plum_pudding_model § Thomson's_1910_scattering_model

At the time of Rutherford's paper, JJ Thomson was the "undisputed world master in the design of atoms". Rutherford needed to compare his new approach to Thomson's. Thomson's model, presented in 1910, modelled the electron collisions with hyperbolic orbits from his 1906 paper combined with a factor for the positive sphere. Multiple resulting small deflections compounded using a random walk.

In his paper Rutherford emphasised that single scattering alone could account for Thomson's results if the positive charge were concentrated in the centre. Rutherford computes the probability of single scattering from a compact charge and demonstrates that it is 3 times larger than Thomson's multiple scattering probability. Rutherford completes his analysis including the effects of density and foil thickness, then concludes that thin foils are governed by single scattering, not multiple scattering.

Later analysis showed Thomson's scattering model could not account for large scattering. The maximum angular deflection from electron scattering or from the positive sphere each come to less than 0.02°; even many such scattering events compounded would result in less than a one degree average deflection and a probability of scattering through 90° of less than one in 10³⁵⁰⁰.

Limitations to Rutherford's scattering formula

Very light nuclei and higher energies

In 1919 Rutherford analyzed alpha particle scattering from hydrogen atoms, 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. Later experiments based on cyclotron acceleration of alpha particles striking heavier nuclei provided data for analysis of interaction between the alpha particle and the nuclear surface. However at energies that push the alpha particles deeper they are strongly absorbed by the nuclei, a more complex interaction.

Quantum mechanics

Rutherford's treatment of alpha particle scattering seems to rely on classical mechanics and yet the particles are of sub-atomic dimensions. However the critical aspects of the theory ultimately rely on conservation of momentum and energy. These concepts apply equally in classical and quantum regimes: the scattering ideas developed by Rutherford apply to subatomic elastic scattering problems like neutron-proton scattering.

Why the plum pudding model was wrong

J. J. Thomson himself didn't study alpha particle scattering, but he did study beta particle scattering. In his 1910 paper "On the Scattering of rapidly moving Electrified Particles", Thomson presented equations that modelled how beta particles scatter in a collision with an atom. Rutherford adapted those equations to alpha particle scattering in his 1911 paper "The Scattering of α and β Particles by Matter and the Structure of the Atom".

Thomson's model relied on electron scattering. In his previous 1904 work Thomson assume the atom had thousands of electrons. Because the electron has only a single charge, the effect of each electron was small, but with multiple scattering the effects could add up. However by 1910 the gold atom was known to have closer to the modern value of 79 electrons.

Deflection by the electrons

Thomson's model for electron scattering was the Coulomb scattering model also used by Rutherford for his atomic centre. However, Thomson worked with electrons colliding with electrons: the particle and the scattering centre have the same mass and charge. Rutherford worked with alpha particles scattering from a concentrated charge having the full mass of the atom. Each collision in Thomson's case has a small effect. He did have more scattering centres, so he had to rely on multiple scattering.

Figure 6

Deflection by the positive sphere

In his 1910 paper, Thomson modeled the scattering an incoming beta particle from positive sphere in the plum pudding model. The same model applies to an alpha particle.

Unlike Rutherfords compact centre, Thomson's model is a sphere of uniformly distributed positive charge (no electrons) with the full radius of the atom. If particle passes just close enough to graze the edge of the sphere, the impact parameter will equal the radius of the atom. At this impact parameter, the particle feels the full force of the 79 electrons in the gold atom, but at a distance 10,000 times larger than the closest approach distance used by Rutherford. Since the force between the particle and the positive sphere scales with the inverse second power, the force in Thomson's model in this case is insignificant.

Figure 4

If the particle passes inside the positive sphere, the force is even less. Inside the sphere the particle will feel only a fraction of the total force.

Figure 5

Cumulative effect

On average the positive sphere and the electrons alike provide very little deflection in a single collision. Thomson's model combined many single-scattering events from the atom's electrons and a positive sphere. Each collision may increase or decrease the total scattering angle. Only very rarely would a series of collisions all line up in the same direction. The result is similar to the standard statistical problem called a random walk. If the average deflection angle of the alpha particle in a single collision with an atom is ${\bar {\theta }}$, then the average deflection after n collisions is

$${\bar {\theta }}_{n}={\bar {\theta }}{\sqrt {n}}$$

The probability that an alpha particle will be deflected by a total of more than 90° after n deflections is given by:

$$e^{-(90/{\bar {\theta }}_{n})^{2}}$$

where e is Euler's number (≈2.71828...). A gold foil with a thickness of 1.5 micrometers would be about 10,000 atoms thick. If the average deflection per atom is 0.008°, the average deflection after 10,000 collisions would be 0.8°. The probability of an alpha particle being deflected by more than 90° will be

$$e^{-(90/0.8)^{2}}\approx e^{-12656}\approx 10^{-5946}$$

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 (100 q_e). The atomic weight of gold was known to be around 197 since early in the 19th century. From an experiment in 1906, Rutherford measured alpha particles to have a charge of 2 q_e and an atomic weight of 4, and alpha particles emitted by radon to have velocity of 1.70×10⁷ m/s. 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 1906, J. J. Thomson measured the elementary charge to be about 3.4×10⁻¹⁰ esu (1.3×10⁻¹⁹ C). In 1909 Robert A. Millikan provided a more accurate measurement of 1.5924×10⁻¹⁹ C, only 0.6% off the current accepted measurement. Jean Perrin in 1909 measured the mass of the hydrogen atom to be 1.43×10⁻²⁷ kg, and if an alpha particle is four times as heavy as that, it would have an absolute mass of 5.72×10⁻²⁷ kg.

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.

See also

• Atomic theory
• Rutherford backscattering spectroscopy
• List of scattering experiments

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External links

• Description of the experiment, from cambridgephysics.org