The nucleus is the tiny, dense core of the atom, containing almost all of its mass. This chapter studies the composition, size, and stability of nuclei, the energy that binds their constituents, and the phenomena of radioactivity, fission, and fusion. Nuclear physics explains both the energy of stars and the release of energy in nuclear reactors.
A nucleus is composed of protons and neutrons, collectively called nucleons. The number of protons determines the atomic number Z, and the total number of nucleons is the mass number A. Nuclear stability depends on the delicate balance between the attractive nuclear force and the repulsive electrostatic force between protons.
Einstein's relation E = mc^2 reveals that mass and energy are equivalent, so the mass of a nucleus is slightly less than the sum of its constituent masses. This mass defect is released as binding energy. This chapter also examines radioactivity, the law of radioactive decay, and the concepts of half-life and mean life, which have applications in dating and medicine.
A nucleus is characterized by its atomic number Z (the number of protons), its neutron number N, and its mass number A = Z + N. Nuclei with the same Z but different A are isotopes (like C-12 and C-14); nuclei with the same A but different Z are isobars (like Ca-40 and Ar-40); and nuclei with the same N but different Z are isotones.
The atomic mass unit is defined as one-twelfth of the mass of a carbon-12 atom:
1 u = 1.66 x 10^-27 kg
The nuclear radius varies with the mass number according to:
R = R0 A^(1/3)
where R0 = 1.2 x 10^-15 m. The volume of the nucleus is therefore proportional to the mass number, meaning all nuclei have nearly the same density, about 2.3 x 10^17 kg/m^3. The nucleus is extremely dense: a sphere of it the size of a cricket ball would weigh about a billion tonnes.
The mass of a nucleus is always less than the sum of the masses of its individual nucleons. This difference is called the mass defect:
Delta M = (Z m_p + N m_n) - M_nucleus
According to Einstein's relation, the mass defect corresponds to the binding energy of the nucleus:
E = Delta M c^2
The binding energy is the energy required to break a nucleus into its constituent nucleons. The binding energy per nucleon measures the stability of a nucleus. It is about 8 MeV per nucleon for most nuclei, with a maximum near iron (A = 56), around 8.75 MeV per nucleon.
The binding energy per nucleon curve shows that: 1. Light nuclei have lower binding energy per nucleon and can release energy by fusion. 2. Very heavy nuclei (like uranium) have lower binding energy per nucleon and can release energy by fission. 3. Iron is the most stable nucleus, with the highest binding energy per nucleon.
Radioactivity is the spontaneous emission of radiation from unstable nuclei. It was discovered by Henri Becquerel in 1896 and studied by Marie and Pierre Curie. There are three types of radiation:
Alpha particles are helium nuclei (He-4), emitted by heavy nuclei. They have low penetrating power (stopped by paper) but high ionizing power. In alpha decay, the nucleus loses Z by 2 and A by 4.
Beta particles are fast electrons (or positrons) emitted when a neutron converts to a proton (or vice versa) inside the nucleus. They have greater penetrating power than alpha particles. In beta-minus decay, Z increases by 1 and A stays the same.
Gamma rays are high-energy electromagnetic waves emitted when a nucleus in an excited state returns to a lower energy state. They have the greatest penetrating power, requiring thick lead or concrete to stop them.
The law of radioactive decay states that the rate of decay is proportional to the number of radioactive nuclei present:
dN/dt = -lambda N
where lambda is the decay constant. Integrating gives:
N = N0 e^(-lambda t)
where N0 is the initial number of nuclei. The half-life, the time for half the nuclei to decay, is related to the decay constant by:
T_1/2 = ln 2 / lambda = 0.693 / lambda
The mean life (average lifetime) of a radioactive nucleus is:
tau = 1 / lambda
Radioactive decay is a random, statistical process; it is not possible to predict which nucleus will decay next, only the probability per unit time. The decay constant and half-life are unique properties of each radioisotope, making radioactivity useful for dating ancient materials.
Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei, releasing a large amount of energy. When a uranium-235 nucleus absorbs a slow neutron, it becomes an unstable uranium-236 nucleus that splits into two lighter nuclei, typically barium and krypton, emitting neutrons and about 200 MeV of energy:
U-235 + n -> Ba-144 + Kr-89 + 3 n + energy
The fission of one nucleus releases several neutrons, which can cause further fissions in a chain reaction. If the reaction is uncontrolled, it produces an explosion (atomic bomb). In a controlled chain reaction, the neutron population is regulated to release energy steadily.
A nuclear reactor controls the chain reaction using: 1. Fuel: U-235 or Pu-239. 2. Moderator: heavy water or graphite, which slows the neutrons to increase fission probability. 3. Control rods: cadmium or boron, which absorb excess neutrons. 4. Coolant: to remove the heat generated.
The energy released in fission per kilogram is about a million times greater than in chemical combustion.
Nuclear fusion is the combining of two light nuclei to form a heavier nucleus, releasing energy. The fusion of hydrogen isotopes:
H-2 + H-3 -> He-4 + n + 17.6 MeV
Fusion releases enormous energy per unit mass, far more than fission, because light nuclei have low binding energy per nucleon and move toward the stable, high-binding-energy region. Fusion requires extremely high temperatures (about 10^7 to 10^8 K) to overcome the Coulomb repulsion between nuclei, so it is called a thermonuclear reaction.
Fusion is the source of the Sun's energy: the Sun fuses hydrogen into helium, releasing the energy that sustains life on Earth. Controlled fusion on Earth remains a challenge, but projects like ITER aim to achieve it. Fusion is attractive because its fuel (deuterium from seawater) is abundant and its products are not dangerously radioactive.
| Quantity | Formula | Value/Remark |
|---|---|---|
| Nuclear radius | R = R0 A^(1/3) | R0 = 1.2 x 10^-15 m |
| Atomic mass unit | 1 u | 1.66 x 10^-27 kg |
| Mass defect | Delta M = (Z mp + N mn) - M | Negative binding |
| Binding energy | E = Delta M c^2 | Per nucleon ~8 MeV |
| Decay law | N = N0 e^(-lambda t) | Statistical process |
| Half-life | T_1/2 = 0.693/lambda | Unique to isotope |
| Mean life | tau = 1/lambda | 1.44 T_1/2 |
| Fission energy | U-235 + n | ~200 MeV per fission |
| Radiation | Nature | Penetrating power | Z, A change |
|---|---|---|---|
| Alpha | He-4 nucleus | Stopped by paper | Z-2, A-4 |
| Beta | Fast electron | Stopped by aluminium | Z+1, A same |
| Gamma | EM wave | Needs lead/concrete | None |
This chapter examined the nucleus - its composition in terms of protons and neutrons, its size R = R0 A^(1/3), and its constant density. The mass defect and binding energy, connected through E = mc^2, determine nuclear stability, with iron being the most stable nucleus. Radioactivity involves alpha, beta, and gamma emission, described by the decay law N = N0 e^(-lambda t) with half-life T_1/2 = 0.693/lambda. Nuclear fission of heavy nuclei releases about 200 MeV per event and is harnessed in reactors through controlled chain reactions, while nuclear fusion of light nuclei powers the Sun and promises abundant clean energy. Together these phenomena demonstrate the enormous energies locked in the atomic nucleus.