Figure references are to the second edition of Modern Physics by Serway, Moses and Moyer (Saunders, 1997)

unless otherwise stated.

Wave mechanics can be applied to the nucleus of an atom, as well as to its surrounding electrons. Since the nucleus is very small (the particles it contains are highly confined), the Heisenberg uncertainty principle tells us that the momentum uncertainly will be large, the energy levels far apart and the energies involved in nuclear reactions correspondingly large (typically MeV).

Our knowledge of nuclear physics started with Becquerel's accidental discovery (1896) that crystals of a uranium salt emit radiation which blackens a photographic emulsion. Two years later, Marie Curie named this emission of radiation from atomic nuclei radioactivity, and less 20 years later Rutherford had used this radiation to investigate properties of the atom and its nucleus (including the nuclear size).

We now know that an atomic nucleus contains Z protons (Z being the proton number of the atom, equal to the atomic number of the corresponding element), each of which has a positive charge (+e), and N uncharged neutrons (N being the neutron number). Together, these two kinds of particles, when present within a nucleus, are called nucleons and their total number A is the mass number of the nucleus, which is closely related to the atomic weight of the corresponding element. Although Z is always the same for a given element, the number of neutrons (and therefore A) is found to vary, giving rise to different isotopes of the element. Hydrogen normally has Z=1, A=1 but can be made with A=2 (deuterium) and A=3 (tritium). Natural carbon consists mainly of the A=12 isotope (carbon-12) but contains 1.1% of carbon-13, so the atomic weight of carbon is actually 12.011, which represents a weighted average over all the isotopes present.

By definition, the mass of the carbon-12 nucleus is 12u, where u is the atomic mass unit (amu), equal to 1.66 x 10^-27 kg. The mass of an isolated proton has been measured to be 1.0073u and that of a neutron is 1.0087u (about 0.14% larger). The fact that C-12 nucleus has a mass slightly less than that of its 12 constituent particles indicates that some mass is lost (as energy, E=mc^2) when the particles are combined into a nucleus.

We have seen how Rutherford estimated the radius of a silver nucleus to be 2 x 10^-14 m, by observing the angular dependence of alpha-particle scattering. This radius can also be represented as 20 fm, where 1fm = 10^-15m, the femtometer or fermi unit. Rutherford thought the neutron was a combination of two particles: a proton and an electron, since electrons are emitted from nuclei in beta-decay reactions. However, we now know that electrons are created inside the nucleus only at the moment of beta-decay. If they were to exist there permanently, the Heisenberg uncertainty principle would predict their momentum to be at least 1^-20 kg m/s^2, which corresponds to a kinetic energy of about 20 MeV, considerably higher than the energies of the electrons emitted in beta-decay.

Later, the British physicist James Chadwick (a student of Rutherford) discovered the neutron as an elementary particle and in 1935 he was awarded a Nobel prize for this discovery. The neutron was found to a magnetic moment, much smaller than that of an electron but comparable in magnitude to that of a proton. This magnetic moment is sometimes thought of as resulting from the particle spinning about its own axis, but is actually a quantum-mechanical effect characterized by a spin quantum number s. Although the neutron, proton and electron all have s = 1/2 , the magnetic moment of the electron is larger because of its smaller mass.

Measurements on a large number of elements has shown that their nuclear radii are given by the formula

r = r₀A^(1/3)

where r₀ = 1.2 x 10^-15 m . This implies that the average density of a nucleus is A u / (4/3)(pi)r^3 = (3/4pi) u / (r₀)^3 , which is independent of the mass number A . Therefore the nucleus can be imagined as a cluster of tightly-packed nucleons (Fig.13.3), each nucleon having a radius which is independent of its environment. Substituting values for r₀ and for the atomic mass unit u , the nuclear density turns out to be 2.3 x 10^17 kg/m^3 , a factor of 2.3 x 10^14 larger than that of water.

Since its protons are all positively charged, a nucleus would fly apart due to electrostatic repulsion were it not for the existence of attractive nuclear forces between the nucleons. Even so, not all nuclei are stable; those which are lie within a broad band on an N-Z diagram (Fig.13.4). For low Z, this band is centred on the line N=Z, but it curves upwards as Z increases, so the heaviest stable elements have roughly 1.5 times as many neutrons as protons. This curvature is due to the electrostatic repulsion of the protons, requiring more neutrons (which provide attractive nuclear force) as Z increases. Above Z=83, no nuclei are permanently stable.

One way of specifying nuclear stability is in terms of its binding energy Eb , the energy which would have to be supplied to completely separate the component nucleons. If the binding energy per nucleon (Eb/A) is plotted against mass number A (Fig.13.10), the curve rises rapidly (with a few oscillations) for low values of A but then flattens off and starts to decrease. The maximum in Eb/A occurs around A=56 , corresponding to the most stable case (the element iron). In principle, nuclei of heavier elements can release energy by splitting apart (nuclear fission) while those of lighter elements can release energy by joining together (nuclear fusion). Fission of uranium nuclei formed the basis of two nuclear bombs developed as a result of the Manhattan project and dropped onto Japanese cities in 1945; it also provides the energy for generating electricity in all current nuclear reactors. Fusion of hydrogen into helium accounts for the energy production in the sun and other stars. Fusion is also used in thermonuclear weapons (the "hydrogen" bomb) and is the basis of continuing attempts to provide electrical power in a thermonuclear reactor.

The binding energy Eb also represents the decreased mass of a nucleus compared to the combined mass of its constituent particles. For example, adding a neutron to a hydrogen atom would produce an atom of deuterium, whose mass is 2.0141u whereas the mass of the H-atom and neutron is 2.0165u. The decrease in mass is 0.0024u, corresponding to a binding energy of 0.0024u/c^2 = 2.2 MeV.

Various models have been proposed to account for the properties of nuclei, including the Eb/A versus A curve. One of these is the liquid drop model, based on the analogy with a spherical drop of liquid, according to which the binding energy consists of three terms. The first is a positive term representing the cohesive volume energydue to the nuclear forces between nearest-neighbour nucleons, which is proportional to the mass number A. The second term is negative and represents the loss of cohesive energy (proportional to to A^2/3) due to the fact that nucleons located at the surface of the nucleus have fewer nearest neighbours. The third term is also negative and represents the electrostatic energy of repulsion between pairs of protons. When these terms are combined, the binding energy per nucleon approximates the form of the actual curve; see Fig. 11.15 (Beiser, Concepts of Modern Physics, p.402). The liquid drop model also provides an explanation and mental picture of how energy can be released when a heavy nucleus undergoes fission; see Fig.13.12.

Another nuclear model is the shell model, based on analogy with the groupings of electrons (electron shells) which surround an atomic nucleus in a neutral atom. Based on interaction of the spins of the nucleons, this model is able to explain the initial oscillations in the binding energy curve (Fig.13.10) and the fact that the most stable nuclei have an even number of nucleons, with N or Z equal to one of a set of so-called magic numbers: 2, 8, 20, 28, 50, 82 and 126.

Some unstable nuclei (those of radioisotopes) emit radiation, which can fall into one of three categories. Alpha-decay results in the emission of positively-charged alpha particles, now known to be helium nuclei containing two protons and two neutrons. Beta-decay gives rise to beta-rays, identified by J.J. Thomson as negatively charged electrons, or in some cases to positron emission. Gamma-decay results in the emission of gamma rays (energetic photons). The number of particles emitted per second decreases exponentially with time, falling to half the original value in a time known as the half-life of the radioisotope.

The half-life involved in alpha-decay varies enormously, from less than 1 microsecond to more than 10^10 years (approximately the age of the universe), yet the energy of the alpha particles covers a relatively narrow range: 5MeV to 9 MeV. An explanation for this surprising observation was provided by George Gamow in 1928. Two protons and two neutrons are imagined to cluster together inside the nucleus to form an alpha particle whose energy is shown (as a function of distance from the centre of the nucleus) in Fig.13.17. Inside the nucleus, the energy is negative as a result of the attractive nuclear forces; outside the nucleus the nuclear forces rapidly become negligible, leaving a positive potential energy due to electrostatic repulsion which decreases with increasing separation between the alpha particle and nucleus. The alpha particle is therefore held inside the nucleus by a potential barrier. However, this barrier is thin enough to allow the particle to escape by wave-mechanical tunnelling, at a rate which is very sensitive to the width of the potential barrier since the alpha-particle wavefunction falls off exponentially in the barrier region where the particle would be classically forbidden. Therefore the large variation in escape rate (and half-life) becomes understandable.

In beta-decay, a neutron inside the nucleus changes into a proton (or vice versa in the case of positron emission) and an electron (or positron) is created out of the rest energy of the decaying nucleus. Experimentally, it is found that the kinetic energies of the escaping beta-particles cover a large range (from zero to some value Kmax; see Fig.13.19), which is hard to understand since the parent nuclei all have an identical mass. This variable kinetic energy also appears to violate the conservation of linear momentum. Moreover, an electron (or positron) has a spin of 1/2, as do the proton and neutron, so angular momentum is not conserved. In 1930, Wolfgang Pauli provided an explanation: an undetected particle must be emitted, in order to balance the energy and momentum equations. The particle was named the neutrino (since it must be uncharged and have little or no mass) and was eventually detected in 1956. The neutrino has a spin of s = 1/2 and interacts very weakly with matter (making it hard to detect). Its rest energy is known to be less than 7 eV and may possibly be zero.

The beta-decay of carbon-14 nuclei forms the basis of carbon dating of biological samples. Living organisms all contain a small fraction (1.3 x 10^-12) of C-14 relative to C-12, the same fraction as for C atoms in carbon dioxide in the atmosphere since living organisms continuously exchange carbon dioxide with their surroundings. Upon death of the organism, the C-14/C-12 ratio falls because C-14 is radioactive with a half-life of 5730 years. Measurement of the C14/C-12 ratio in wood, charcoal, bone or shell samples therefore can therefore provide the age (since death) within the range 1000 - 25000 years. A famous example is the dating of the Dead Sea Scrolls, manuscripts which were found to have an age of 1950 years.

Gamma-decay occurs when a nucleus undergoing radioactive decay is left in an excited state (analogous to the excited state of an atom) and can therefore decay further, to a lower-energy (e.g. ground) state, by the emission of a high-energy photon. The photon energy is typically in the range 1 MeV to 1000 MeV (1 GeV). Unstable isotopes often decay via a whole series of nuclear reactions, each of which may involve alpha, beta or gamma decay.