# Geiger–Marsden experiment

A replica of one of Geiger and Marsden's apparatus

The Geiger–Marsden experiments (also called the Rutherford gold foil experiment) were a landmark series of experiments by which scientists discovered that every atom contains a nucleus where its positive charge and most of its mass is concentrated. They deduced this by observing how alpha particles are scattered when they strike a thin metal foil. The experiments were performed between 1908 and 1913 by Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester.

## Summary

### Contemporary theories of atomic structure

The popular theory of atomic structure at the time of Rutherford's experiment was the "plum pudding model". This model was developed in 1904 by J. J. Thomson, the scientist who discovered the electron a few years earlier. This theory held that the negatively-charged electrons in an atom are distributed in a uniform sea of positive charge like plums in a bowl of pudding. A competing theory was proposed by Hantaro Nagaoka.[1][2] Nagaoka rejected Thomson's model on the grounds that opposing charges cannot penetrate each other. He proposed instead that the positive charge of the atom is concentrated in a nucleus, with the electrons orbiting it like the rings around Saturn.

### Implications of the plum pudding model

An alpha particle is a sub-microscopic, positively-charged particle of matter. Thomson's model of the atom predicted that if an alpha particle were to come near an atom, its trajectory would be deflected by only a negligible amount because the electric fields within the atom are too weak to affect it much. Both the negative and positive charges within the Thomson atom are spread out over the atom's entire volume. According to Coulomb's Law, the less concentrated an electric charge is, the weaker its electric field at its surface will be.[3][4]

As a worked example, consider an alpha particle passing tangentially to a gold atom, where it will experience the electric field at its strongest and thus experience the maximum deflection θ. Since the electrons are very light compared to the alpha particle, we can neglect their influence entirely and instead see the atom as a heavy sphere of positive charge.

Qn = positive charge of gold atom = 79 e = 1.26558×10−17 C
Qα = charge of an alpha particle = e = 3.204×10−19 C
r = radius of a gold atom = 1.44×10−10 m
vα = velocity of an alpha particle = 1.53×107 m/s
mα = mass of an alpha particle = 6.64×10−27 kg
k = Coulomb's constant = 8.98×109 N·m2/C2

Using classical physics, we can approximate the alpha particle's lateral change in momentum Δp using the impulse of force relationship and the Coulomb force expression:

$\Delta p = F \Delta t = k \cdot \frac {Q_\alpha Q_n}{r^2} \cdot \frac {2r}{v_\alpha}$
$\theta \approx \frac {\Delta p}{p} < k \cdot \frac {2Q_\alpha Q_n}{m_\alpha r v_\alpha^2} = 8.98 \cdot 10^9 \times \frac {2 \times 3.204 \cdot 10^{-19} \times 1.26558 \cdot 10^{-17}}{6.64 \cdot 10^{-27} \times 1.44 \cdot 10^{-10} \times (1.53 \cdot 10^7)^2}$
$\theta < 0.000325~\mathrm{rad}~(\mathrm{or}~0.0186^\circ)$

We can see that at most the alpha particle would be deflected by a small fraction of a degree.

### 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.

At Rutherford's behest, Geiger and Marsden performed a series of experiments where they pointed a beam of alpha particles at a thin foil of metal and measured the scattering pattern by the use of a fluorescent screen. They found that atoms could deflect an alpha particle by as much as 150°. This should have been impossible according to Thomson's model. Obviously, those particles had encountered an electrostatic force far greater than Thomson's model suggested they would, which in turn implied that the atom's positive charge was concentrated in a much tinier volume than Thomson imagined.[5]

When Geiger and Marsden shot alpha particles at their metal foils, they noticed only a tiny fraction of the alpha particles were deflected by more than 90°. Most just flew straight through the foil. This suggested that those tiny spheres of intense positive charge were separated by vast gulfs of empty space.[5] Imagine you are standing on the edge of a copse of trees with a large bag full of tennis balls. If you were to blindly throw tennis balls at the trees, you would notice that most of the balls would fly through hitting nothing, while a few would strike tree trunks and bounce off in all directions. This analogy illustrates what Rutherford saw in the scattering pattern of the alpha particles. Most particles went straight through the metal foil because its matter was mostly empty space, but a few had "struck" some small but strong obstacle: the nuclei of the atoms.

Rutherford saw no option but to dismiss Thomson's model of the atom, and instead propose a model where the atom consisted of mostly empty space, with all its positive charge concentrated in its center in a very tiny volume, surrounded by a cloud of electrons.

## Timeline of events

### Background

Ernest Rutherford was a physics professor at the University of Manchester. He had already received numerous honors 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 promising young German physicist named Hans Geiger, and was so impressed that he asked Geiger to stay and help him with his research.[6] 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 himself 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 even with 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.[6]

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 didn't 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. Rutherford asked Geiger to investigate just how much matter could scatter alpha rays.[7]

The experiments they designed involved bombarding a metal foil with alpha particles to observe how the foil scattered them. Since alpha particles are so tiny, they used a fluorescent screen to measure their trajectories. Each impact of an alpha particle on the screen produced a tiny flash of light. To count these tiny flashes of light, Geiger and Marsden had to work in a darkened lab, peering through a microscope for hours on end.[8] Rutherford lacked the endurance for this work, which is why he left it to his younger colleagues.[9] 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 ductile.[10] As a source of alpha particles, Rutherford's substance of choice was radium and its decay products. Radium is a substance several million times more radioactive than uranium. Because only a very tiny fraction of the alpha particles would experience great angles of deflection, a truly prodigious emitter of alpha particles was needed to obtain statistically meaningful results.

### 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[11] describes the following experiment. Geiger 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. When the air was pumped out of the tube and the alpha particles were unobstructed, they left a neat and tight image on the screen that corresponded to the shape of the slit. When there was some air in the tube, 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°.

### The 1909 experiment

In these very simple 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,[12] Geiger and Marsden described their eponymous 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), beneath which they placed a fluorescent screen (S). The tube was held above the plate, such that the alpha particles it emitted could not directly strike the screen. They noticed a few scintillations on the screen—this was because some alpha particles could circumvent the lead plate by bouncing off air molecules (the experiment was not done in a vacuum). 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 that is indeed what they saw. Counting the scintillations, they noticed 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 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 8000).[12]

### 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[13] by Geiger describes an experiment by which he sought to measure how the most probable angle through which an a-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. From the measurements he took, Geiger found that the most probable angle of deflection increases with the thickness of the material, is proportional to the atomic mass of the substance, and decreases with the velocity of the alpha particles, and that in any case the probability that a particle will be deflected by more than 90° is vanishingly small.

### Rutherford's mathematical model

Considering the results of the above experiments, Rutherford published a landmark paper in 1911 where he proposed the existence of a strong electrical charge concentrated in a very small volume at the center of the atom.[14] Rutherford developed a mathematical equation that modeled how the foil should scatter the alpha particles if all the positive charge was concentrated in a single point at the center of an atom.

$s = \frac {Xnt\csc^4\tfrac {\phi}{2}}{16r^2} \cdot (\frac {2Q_n Q_{\alpha}}{mv^2})^2$

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
Qn = positive charge of the atomic nucleus
Qα = positive charge of the alpha particles
m = mass of an alpha particle
v = velocity of the alpha particles

### The 1913 experiment

In a 1913 paper, 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. csc4Φ/2
2. thickness of foil t
3. magnitude of central charge Qn
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. The microscope and screen 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.[15]

To test how the scattering varied with the angle of deflection (ie 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 the metal foil and the radiation source (containing radon), mounted on a detached column which allowed the cylinder to rotate independently. A microscope with its objective lens covered by a fluorescent zinc sulfide screen 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°.[15] 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.

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.[15]

Geiger and Marsden then tested how the scattering varied with the thickness of the foil (ie if s ∝ t). They constructed a disc with six holes drilled in it. The holes were covered with metal foil of varying thickness, or none for the sake of control. This disc was then sealed in a brass ring between two glass plates. The disc could be rotated by means of a rod to bring each window in front of the alpha particle source. On the rear glass pane was a zinc sulfide screen. 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.

Geiger and Marsden reused the above apparatus to measure how the scattering pattern varied with the square of the nuclear charge (ie if s ∝ Qn2). Geiger and Marsden didn't 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 ∝ Qn2.

Finally, Geiger and Marsden tested how the scattering varied with the velocity of the alpha particles (ie 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.

## Legacy

When Geiger reported to Rutherford that he had spotted alpha particles being strongly deflected, Rutherford was astounded. In a lecture Rutherford delivered at Cambridge University, he said:

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.

—Ernest Rutherford[16]

Accolades soon flooded in. Hantaro Nagaoka, who had once proposed a Saturnian model of the atom, wrote to Rutherford from Tokyo in 1911: "Congratulations on the simpleness of the apparatus you employ and the brilliant results you obtained". The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier.[9] By his discovery of the nucleus, Rutherford began the new science of nuclear physics.