Figure references are to the second edition of Modern Physics by Serway, Moses and Moyer (Saunders, 1989).

Although some ancient Greeks (such as Democritus) postulated the existence of atoms (units of matter which could not be subdivided), concrete evidence for their existence did not develop until the 19th century. The first direct evidence came from observations of the Brownian motion.

Other evidence came from chemistry, such as the experiments of Faraday on electrolysis (1833). Faraday's law states that when a current is passed through a solution or through a molten electrolyte, the mass of a particular element deposited at the cathode or anode is proportional to the electrical charge which has flowed around the circuit (current times time) and to the atomic weight of that element, but inversely proportional to the valence of the element. Although its interpretation was not clear at the time, Faraday's law reflects the fact that electricity passes through an electrolyte in the form of ions whose mass is proportional to the atomic weight and whose charge is equal to the valence, which represents the number of electrons which have been removed from the neutral atom.

The properties of electrons were investigated by J.J.Thomson at the Cavendish Laboratory in 1897 (Figs. 3.2 - 3.4). An electrical discharge in a low-pressure gas was known to produce cathode rays, which could cause an object in their path to emit light (fluoresce), but it was not known if these rays consisted of waves or material particles. Thomson passed a beam of cathode rays through a uniform electric field E (between two parallel plates) and a magnetic field B (produced by an electromagnet) acting over the same region of space (see Fig. 3.3).

The two fields were perpendicular to the beam and to each other, such that the magnetic and electrical forces were both perpendicular to the beam but opposite in direction. By adjusting the strength of one of the fields, so as to produce zero net deflection as observed on a fluorescent screen, he achieved the condition:

e E + B e Vx = 0

and so was able to measure the speed Vx of the particles in the beam (travelling in the x-direction).

With the magnetic field turned off (Fig. 3.5), the electric field produced a force (and therefore an acceleration a) on each particle over a distance L, deflecting the beam through an angle q which could be measured. The magnitude Vy of the velocity acquired in a direction parallel to the field (i.e. perpendicular to the beam) can be calculated from

Vy = a t = (eE/m) (L/Vx) , so that: tan (theta) = Vy / v = (L E / Vx^2) (e/m)

By measuring (theta) , L and E (=V/d where V is the voltage between the plates, separation d) the charge-to-mass ratio e/m of the particles could be found.

The value of e/m was found to be independent of the electrodes and the gas in the discharge tube, suggesting that the cathode rays consisted of particles which were a constituent of all matter. The value of e/m was about 2000 times larger than the charge/mass ratio measured (by electrolysis of water) for hydrogen ions, indicating a particle much smaller in mass than the smallest atom.

The electronic charge e was first determined by Robert Millikan (in 1909) by observing the motion of individual drops of oil in an vapour, ionized with a radioactive source; see Fig. 3.7. A drop acquires a terminal velocity as a result of a balance of the viscous(air-resistance), gravitational and (with an electric field applied) electrostatic forces. Applying a formula for the viscous force on a spherical drop and measuring the velocity with an without applied field, the charge on the drop can be calculated.

Although atoms were originally thought of as being indivisible, evidence began to accumulate (towards the end of the 19th century) that the atom contains component parts: electrons (which can be emitted as cathode rays) and a balancing positive charge (to give overall neutrality and prevent the atom exploding from the electrostatic repulsion of all the negatively charged electrons).

In 1898, J.J. Thomson proposed that the electrons are embedded (like plums in a pudding) in a sphere of uniformly distributed positive charge. Other physicists came up with other ideas, for example the positive charge might be concentrated in a central nucleus with the electrons orbiting around it, analagous to the solar system, except that the attractive forces would be electrostatic rather than gravitational.

Experiments conducted between 1909 and 1924, supervised by Ernest Rutherford but actually carried out by two of his students (Hans Geiger and Ernst Marsden), provided the evidence necessary to choose between these models. The experiments arose out of investigations of radioactive materials, discovered by Becquerel in 1896 when he accidentally left wrapped photographic plates in a drawer together with some uranium-salt crystals (he shared a 1903 Nobel prize with Pierre and Marie Curie).

The apparatus was quite simple: a radioactive source (such as radon gas) emitted alpha particles, which have a charge of +2e and a mass about four times that of the hydrogen atom (they are actually helium-atom nuclei). A parallel beam of these particles (collimated by a lead tube) was directed towards a thin foil of a metal such as gold, mounted in a vacuum chamber. Some of the alpha particles passed through the foil without measurable deviation but some were scattered through appreciable angles (j in Fig.3.9). The angular distribution of the scattered a -particles was measured by mounting a fluorescent screen (a slab of glass coated with fine zinc suphide particles) at the entrance of a low-power telescope, which could be rotated about the scattering point; see Fig. 3.9. With a dark-adapted eye, single a -particles hitting the screen could be detected as a small flash of light seen in the telescope. Most of the alpha particles were deflected through small angles (of the order of 1 degree) but a small fraction were scattered through larger angles, including some backscattered through angles exceeding 90 degrees = p /2 radians; see Fig. 3.11.

Rutherford realized that the existence of large-angle scattering ruled out the Thomson (plum-pudding) model; the relatively heavy a -particles would not be turned around by much lighter electrons or by the combined mass of a gold atom if this mass were distributed over the whole atomic volume. On the other hand, if the positive charge and most of the mass of a gold atom were concentrated in a central nucleus, its electrostatic repulsion would repel incoming a -particles and deflect some of them through angles as large as 180 degrees (see Fig. 3.10) without absorbing much of the a -particle's energy (i.e. the "collision" will be elastic). This scattering from the electrostatic field of the nucleus is now known as Rutherford scattering. Note that if the nucleus is small compared to the whole atom (it typically occupies less than 1 part in 10^15 of the volume!), the probability of high-angle a -scattering will be very small, in agreement with the measured angular distribution (note the logarithmic vertical scale in Fig. 3.11).

Based on his nuclear model of the atom, Rutherford was able to calculate an expression for the angular distribution of the a -particle scattering. He needed to know the magnitide F of the force on an a -particle when it is a distance r from the centre of a nucleus; this is given by Coulomb's law:

F = k (+2e)(Ze)/r^2

where k is the Coulomb constant and (Ze) is the nuclear charge, Z being the atomic number of the atoms in the foil. Applying Newton's second law (and conservation of momentum and energy) to the two-particle interaction led to the following expression for the number n of alpha particles detected at a scattering angle (phi) :

n = C ( N Z^2 / K^2 ) / [sin (phi)/2]^4

where K is the kinetic energy of the alpha -particles and C is a parameter which depends on the strength of the alpha -particle source and the geometry of the particle detector; N is the number of nuclei per unit area of the foil, equal to (rho) t / (A u) where t and (rho) are the foil thickness and density, while A represents atomic weight and u is the atomic-mass unit. Careful experiments by Geiger and Marsden, published in 1913 (Philosophic Magazine 25, p.605), confirmed the t, K and phi dependence implied by this formula.

At that time, atomic numbers for different elements were not known; however, Rutherford was able to fit the results obtained from different foils to his formula by trying different values of Z ; see Fig. 3.11. Thereby, he was able to measure Z , while the relatively good degree of fit provided additional confirmation of his theory.

Furthermore, Rutherford realized that the remarkable success of his formula provided information about the size of the nucleus. He argued that if an alpha actually reached the nucleus, the latter would be "deformed" - in other words the force law would depart from the Coulomb's law expression and the angular dependence of scattering would change. A particle which gets closest to a nucleus will be one which directly approaches its centre and is deflected through 180 degrees (as in Fig. 3.10). At the moment of its closest approach, this particle will be momentarily stationary, having exchanged all of its kinetic energy K into electrostatic energy of repulsion, so that:

K = k (Ze) (2e) / r

Smaller values of r will be possible by choosing foils of lower atomic number and alpha particles of higher kinetic energy. The experimenters therefore worked with aluminum foils and other radioactive sources which provided more penetrating radiation, looking for evidence for departures from Rutherford's formula in the measured angular distributions. In 1919, they were able to detect a departure from the formula for 7.7MeV alpha-particles (K = 7.7 x 10^6 / 1.6 x 10^-19 Joule) scattered from a foil of aluminum (Z=13), allowing the radius of the Al nucleus to be estimated as: r = 2 k Z e^2 / K = 4.9 x 10^-15 m. Since the radius of an atom is of the order of 10^-10 m, the atom is seen to be mainly empty space.

Despite the success of the nuclear atomic model, it raised several questions. For example, the atomic number (a measure of the nuclear charge) of an element turned out to be more than a factor two lower than its atomic weight (a measure of the nuclear mass mass). Rutherford speculated that the nucleus might contain electrons, which would neutralize some of its positive charge without adding appreciable mass. This seemed plausible at the time, because certain types of radioactive materials emit high-energy electrons (beta- rather than alpha-decay) which might come from the nucleus.

Rutherford thought that the presence of electrons might explain another problem with the nuclear model: why does the nucleus not fly apart because of electrostatic repulsion of the positive particles? Perhaps the electrons act as a kind of glue which holds the nucleus together ? Only later did it become clear that, at the close separations involved in the nucleus, its component particles exhibit strong nuclear forces which completely overwhelm the electrostatic repulsion and which represent an entirely different kind of interaction.

Probably the most serious problem with the planetary model is that an orbiting electron has a centripetal acceleration and (according to Maxwell's theory of electromagnetism) ought to lose energy by emitting electromagnetic radiation at a frequency equal to that of the orbital motion (the reciprocal of the orbital period). This radiated energy would be at the expense of the electrostatic potential energy of the electron, which would become more negative - implying that the electron approaches closer to the nucleus and experiences an increased electrostatic force. This increased force implies an increased centripetal acceleration and a higher angular velocity of the orbiting electron; the frequency of the emitted radiation would increase and the electron would spiral into the nucleus, as indicated in Fig. 3.20. Calculations showed that this process should happen in a small fraction of a second; in other words, the atom should not be stable ! The problem was not solved by Rutherford; it took the genius of Niels Bohr to propose a solution.