Contemporary theories of atomic structure

The plum pudding model of the atom, as envisioned by Thomson

The prevailing theory of atomic structure at the time of Rutherford's experiments was the "plum pudding model". This model was devised by J. J. Thomson. Thomson had discovered the electron and proposed that every atom was a sphere of positive charge throughout which the electrons were distributed, a bit like raisins in a Christmas pudding. The existence of protons and neutrons was unknown at this time. This model was based entirely on classical (Newtonian) physics; the current accepted model uses quantum mechanics.

Thomson's model was not universally accepted even before Rutherford's experiments. Thomson himself was never able to develop a complete and stable model of his concept. Japanese scientist Hantaro Nagaoka rejected Thomson's model on the grounds that opposing charges cannot penetrate each other.^[1] He proposed instead that electrons orbit the positive charge like the rings around Saturn.^[2]

An alpha particle is a sub-microscopic, positively charged particle of matter that is spontaneously emitted from certain radioactive elements. Rutherford discovered their existence and deduced that they were essentially helium atoms without electrons, though nobody knew about protons and neutrons at the time.^[3]

According to Thomson's model, if an alpha particle were to collide with an atom, it would just fly straight through, its path being deflected by at most a fraction of a degree. At the atomic scale, the concept of "solid matter" is meaningless. The Thomson atom is a sphere of electric charge anchored in space by its mass, with mass by definition being resistance to acceleration. Thus the alpha particle will not bounce off the atom like a tennis ball hitting a basketball, but will pass right through if the atom's electric fields are weak enough to permit it. Thomson's model predicted that the electric fields in an atom are too weak to affect a passing alpha particle much, given how fast and heavy alpha particles are. Both the negative and positive charges within the Thomson atom are spread out over the atom's entire volume, and Rutherford had calculated that this volume was too large for strong deflection to happen. According to Coulomb's law, the less concentrated a sphere of electric charge is, the weaker its electric field at its surface will be.^[4]

Consider an alpha particle passing along the edge of a gold atom, where it will experience the electric field at its strongest and thus experience the maximum deflection angle θ. Since the electrons are very light compared to the alpha particle, their influence can be neglected and the atom can be modeled as a heavy sphere of positive charge.

Using classical physics, the alpha particle's lateral change in momentum Δp_y can be approximated using the impulse of force relationship and the Coulomb force expression:^[5]

$${\displaystyle \Delta p_{y}={\bar {F}}t<k{\frac {q_{\alpha }q_{g}}{r^{2}}}\cdot {\frac {2r}{v}}}$$

where

• q_g = positive charge of gold atom = 79 e = 1.266×10⁻¹⁷ C
• q_α = charge of alpha particle = 2 e = 3.204×10⁻¹⁹ C
• r = radius of gold atom = 1.44×10⁻¹⁰ m
• v = velocity of alpha particle = 1.53×10⁷ m/s
• m = mass of alpha particle = 6.645×10⁻²⁷ kg
• k = Coulomb constant = 8.987×10⁹ N·m²/C²

This gives the deflection angle

$${\displaystyle \tan \theta ={\frac {\Delta p_{y}}{p_{x}}}<k{\frac {q_{\alpha }q_{g}}{r^{2}}}\cdot {\frac {2r}{v}}\cdot {\frac {1}{mv}}=0.000325}$$

$${\displaystyle \theta <0.000325~{\text{radians}}~({\text{or}}~0.0186^{\circ })}$$

One could obtain a more accurate estimate for the deflection angle using the impact parameter equation, but this simpler calculation is sufficient to show the order of magnitude. For the alpha particle to be deflected by more than 1 degree, the sphere of positive charge would have to be several orders of magnitude smaller.

Now consider an alpha particle passing through the middle of the same atom.

As the alpha particle approaches the center of the atom, the atom's electric field pushes back on it. If the alpha particle passes the center of the atom, the atom will then start pushing the alpha particle forwards instead of back.

Let x be the distance between the alpha particle and the center of the atom. The amount of work that the atom exerts on the alpha particle up to the point that they make contact (ie when x ≥ r) will at most be

$${\displaystyle {\bar {F}}\cdot \mathrm {d} x=\int _{r}^{\infty }k{\frac {q_{\alpha }q_{g}}{x^{2}}}\cdot \mathrm {d} x=\left[-k{\frac {q_{\alpha }q_{g}}{x}}\right]_{r}^{\infty }=2.53\times 10^{-16}~{\text{Joules}}}$$

The amount of work exerted on the alpha particle as it passes through the atom from the surface to the center (ie when x < r) is given by^[6]

$${\displaystyle \int _{0}^{r}k{\frac {q_{\alpha }q_{g}}{x^{2}}}\cdot {\frac {x^{3}}{r^{3}}}\cdot \mathrm {d} x=\left[k{\frac {q_{\alpha }q_{g}x^{2}}{2r^{3}}}\right]_{0}^{r}=1.27\times 10^{-16}~{\text{Joules}}}$$

Add these two figures together, and the total amount of work exerted on the alpha particle as it approaches the center of the atom is 3.80×10⁻¹⁶ J.

The initial kinetic energy of the alpha particle is given by

$${\displaystyle {\frac {1}{2}}mv^{2}=7.78\times 10^{-13}~{\text{Joules}}}$$

which is several orders of magnitude more than the work exerted by the atom's electric field. The Thomson atom thus cannot bring the alpha particle to a halt, never mind send it flying backwards.

The calculations above use modern measurements, but Rutherford had sufficiently accurate estimates. He assumed that the radius of atoms in general to be on the order of 10⁻¹⁰ m (perhaps after reading Jean Perrin's studies on Brownian motion^[7]) and the positive charge of a gold atom to be about 100 times that of hydrogen (100 e).^[8] He knew that gold has an atomic weight of 197. From an experiment in 1906, Rutherford measured alpha particles to have a charge of 2 e and an atomic weight of 4, and alpha particles emitted by radon to have velocity of 1.70×10⁷ m/s.^[9] Jean Perrin in 1909 measured the mass of hydrogen to be 1.43×10⁻²⁷ kg,^[10] and if alpha particles are four times as heavy as that, they would have an absolute mass of 5.72×10⁻²⁷ kg.

The outcome of the experiments

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.

Between 1908 and 1913, Rutherford, Geiger, and Marsden ran a series of experiments where they pointed a beam of alpha particles at thin foils of various metals and studied the scattering pattern of the alpha particles with the aid of a fluorescent screen. They discovered that the metal foils could scatter some alpha particles in all directions, sometimes more than 90 degrees. This should have been impossible according to Thomson's model; the alpha particles should have all gone straight through. Obviously, those particles had encountered an electrostatic force far greater than Thomson's model suggested they would.^[11]

These bizarre results led Rutherford to conclude in 1911 that the atom is a largely open structure containing a tiny nucleus at its center where its positive charge was concentrated.^[12]

Legacy

Hantaro Nagaoka, who had once 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."^[13] The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier.^[14]

In a lecture delivered on October 15, 1936 at Cambridge University,^[15][16] Rutherford commented on his reaction to the results of the 1909 experiment (see below for details):

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 realized 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.^[17]

According to classical Newtonian physics, Rutherford's model of the atom is impossible. Accelerating charged particles radiate electromagnetic waves, so an electron orbiting an atomic nucleus in theory would spiral into the nucleus as it loses energy. Obviously this was not happening, which meant that the classic laws of physics do not apply at the atomic scale. This eventually led Niels Bohr to incorporate quantum mechanics into the model of the atom. These developments came roughly at the same time Albert Einstein produced his theory of general relativity, which showed that the classic laws of physics do not apply at the cosmic scale either.

Background

Ernest Rutherford

Hans Geiger

Ernest Marsden

Ernest Rutherford was Langworthy Professor of Physics at the Victoria University of Manchester^[18] (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 a German physicist named Hans Geiger, and was so impressed that he asked Geiger to stay and help him with his research.^[19] Ernest Marsden was a physics undergraduate student studying under Geiger.

Alpha particles are tiny, positively charged particles that are spontaneously emitted by certain substances such as uranium and radium. Rutherford had discovered them in 1899. In 1908, he was trying to precisely measure their charge-to-mass ratio. To do this, he first needed to know just how many alpha particles his sample of radium was giving off (after which he would measure their total charge and divide one by the other). Alpha particles are too tiny to be seen with the naked eye or even a microscope, but Rutherford knew that alpha particles ionize air molecules, and if the air is within an electric field, the ions will produce an electric current. On this principle, Rutherford and Geiger designed a simple counting device which consisted of two electrodes in a glass tube. Every alpha particle that passed through the tube would create a pulse of electricity that could be counted. It was an early version of the Geiger counter.^[19]

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 just too heavy to be deflected so strongly (in 1906, he measured alpha particles as having the same weight as helium atoms^[20]). Rutherford asked Geiger to investigate just how much matter could scatter alpha rays.^[21]

The experiments they designed involved bombarding a metal foil with alpha particles to observe how the foil scattered them in relation to their thickness and material. They used a fluorescent 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.^[4] Rutherford lacked the stamina and patience for this work, being in his late 30s, which is why he left it to his younger colleagues.^[22] For the metal foil, they tested a variety of metals, but they preferred gold because they could make the foil very thin, as gold is very malleable.^[23] As a source of alpha particles, Rutherford's substance of choice was radon, a substance several million times more radioactive than uranium.

The 1908 experiment

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 meters in length. At one end of the tube was a quantity of "radium emanation" (R) that served 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 in the tube, and the glowing patch became more diffuse. Geiger then pumped out the air and placed some gold foil over the slit at AA. This too caused the patch of light on the screen to become more spread out. This experiment demonstrated that both air and solid matter could markedly scatter alpha particles. The apparatus, however, could only observe small angles of deflection. Rutherford wanted to know if the alpha particles were being scattered by even larger angles—perhaps larger than 90°.^[24]

The 1909 experiment

In these experiments, alpha particles emitted by a radioactive source (A) were observed bouncing off a metal reflector (R) and onto a fluorescent screen (S) on the other side of a lead plate (P).

In a 1909 paper, On a Diffuse Reflection of the α-Particles,^[25] 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 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 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.^[25]

Geiger and Marsden then wanted to estimate the total number of alpha particles that were being 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 found that only a tiny fraction of the alpha particles that struck the reflector bounced onto the screen (in this case, 1 in 8,000).^[25]

The 1910 experiment

This apparatus was described in 1910 paper by Geiger. It was designed to precisely measure how the scattering varied according to the substance and thickness of the foil.

A 1910 paper^[26] by Geiger, The Scattering of the α-Particles by Matter, describes an experiment by which he sought to measure how the most probable angle through which an alpha particle is deflected varies with the material it passes through, the thickness of said 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 millimeter 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 was 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 could also vary the velocity of the alpha particles by placing extra sheets of mica or aluminium at A.^[26]

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
• the probability that a particle will be deflected by more than 90° is vanishingly small

Rutherford mathematically models the scattering pattern

Considering the results of the above 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 proposed that the atom contains at its center a volume of electric charge that is very small and intense (in fact, Rutherford treats it as a point charge in his calculations).^[8] 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 modeled 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 center of an atom.

$${\displaystyle s={\frac {Xnt\cdot \csc ^{4}{\!\left({\tfrac {\phi }{2}}\right)}}{16r^{2}}}\cdot {\left({\frac {2q_{n}q_{\alpha }}{mv^{2}}}\right)}^{2}}$$

• s = the number of alpha particles falling on unit area at an angle of deflection Φ
• r = distance from point of incidence of α rays on scattering material
• X = total number of particles falling on the scattering material
• n = number of atoms in a unit volume of the material
• t = thickness of the foil
• q_n = positive charge of the atomic nucleus
• q_α = positive charge of the alpha particles
• m = mass of an alpha particle
• v = velocity of the alpha particle

From the scattering data, Rutherford estimated the central charge q_n to be about +100 units (see Rutherford model)

The 1913 experiment

In a 1913 paper, The Laws of Deflexion of α Particles through Large Angles,^[27] Geiger and Marsden describe a series of experiments by which they sought to experimentally verify the above equation that Rutherford developed. 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:

1. csc⁴(Φ/2)
2. thickness of foil t
3. magnitude of the square of central charge Q_n
4. 1/(mv²)²

Their 1913 paper describes four experiments by which they proved each of these four relationships.

This apparatus was described in a 1913 paper by Geiger and Marsden. It was designed to accurately measure the scattering pattern of the alpha particles produced by the metal foil (F). The microscope (M) and screen (S) were affixed to a rotating cylinder and could be moved a full circle around the foil so that they could count scintillations from every angle.^[27]

To test how the scattering varied with the angle of deflection (i.e. if s ∝ csc⁴(Φ/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. 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 csc⁴(Φ/2).^[27]

This apparatus was used to measure how the alpha particle scattering pattern varied in relation to the thickness of the foil, the atomic weight of the material, and the velocity of the alpha particles. The rotating disc in the center had six holes which could be covered with foil.^[27]

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 zinc sulfide screen was indeed proportional to the thickness as long as said thickness was small.^[27]

Geiger and Marsden reused the above apparatus to measure how the scattering pattern varied with the square of the nuclear charge (i.e. if s ∝ Q_n²). 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 aluminum. 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 more or less the same. Thus they proved that s ∝ Q_n².^[27]

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 again, 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, that the number of scinitillations was indeed proportional to 1/v⁴.^[27]

Rutherford determines the nucleus is positively charged

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.^[28] In a 1913 paper,^[29] 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 an experiment where they shot a beam of alpha particles through hydrogen, the alpha particles knocked the hydrogen nuclei forwards in the direction of the beam, not backwards. In an experiment where they shot alpha particles through nitrogen, he discovered that the alpha particles knocked hydrogen nuclei (i.e. protons) out of the nitrogen nuclei.^[28]