Chapter 13: Nuclei | Summary

Through electron scattering experiments, it has been established that nuclei are approximately spherical. The nuclear radius R is related to the mass number A by the empirical formula R = R₀A^(1/3), where R₀ = 1.2 × 10^-15 m (1.2 fm). This formula implies that the volume of a nucleus is directly proportional to A.

Since volume is proportional to A and mass is also proportional to A (each nucleon has nearly the same mass), the nuclear matter density is constant and independent of A for all nuclei. This density is approximately 2.3 × 10^17 kg/m³, which is enormously large compared to ordinary matter. This is consistent with the fact that most of the atom is empty space.

Einstein's mass-energy relation E = mc² established that mass is a form of energy. The measured mass of a nucleus is always less than the sum of the masses of its individual protons and neutrons. This difference, called the mass defect ΔM, is given by ΔM = [Zmp + (A–Z)mn] – M. The energy equivalent of this mass defect, Eb = ΔMc², is the binding energy of the nucleus — the energy required to completely separate all nucleons.

The binding energy per nucleon Ebn = Eb/A is a measure of nuclear stability. A plot of Ebn versus A shows that it is nearly constant (~8 MeV) for 30 < A < 170, peaks around A = 56 (iron) at about 8.75 MeV, and is lower for very light and very heavy nuclei. This curve is key to understanding both fission (heavy nucleus splits) and fusion (light nuclei combine), both of which release energy by moving toward more stable, higher-Ebn configurations.

The nuclear force is the strong attractive force that holds protons and neutrons together inside the nucleus, overcoming the electrostatic repulsion between protons. It is far stronger than both the Coulomb force and gravitational force. The nuclear force is short-ranged — it drops rapidly to zero for separations greater than a few femtometres.

At a separation of about 0.8 fm, the nuclear potential energy is at a minimum. The force is attractive for distances greater than 0.8 fm and becomes strongly repulsive for distances less than 0.8 fm. Importantly, the nuclear force between neutron-neutron, proton-neutron, and proton-proton pairs is approximately the same — it is charge-independent. There is no simple mathematical expression for the nuclear force unlike Coulomb's or Newton's law.

Radioactivity, discovered accidentally by Henri Becquerel in 1896, is a nuclear phenomenon in which an unstable nucleus undergoes spontaneous decay by emitting radiation. Three types of radioactive decay occur in nature: alpha (α) decay, where a helium nucleus is emitted; beta (β) decay, where electrons or positrons are emitted; and gamma (γ) decay, where high-energy photons are emitted.

Radioactivity is an indication of nuclear instability. For light nuclei, a neutron-to-proton ratio of about 1:1 is needed for stability, while for heavy nuclei this ratio increases to about 3:2. Nuclei that deviate from the stability curve — either with excess neutrons or excess protons — are unstable. Only about 10% of all known isotopes are stable; the rest are radioactive.

Nuclear fission is the process in which a heavy nucleus (such as U-235) is bombarded by a neutron and splits into two intermediate-mass fragments along with 2–3 neutrons and a release of approximately 200 MeV of energy. This enormous energy output — about a million times that of chemical reactions for the same mass — is the basis of nuclear reactors (controlled fission) and atomic bombs (uncontrolled fission). The Q-value of the fission reaction is derived from the increase in binding energy per nucleon from ~7.6 MeV (for A = 240) to ~8.5 MeV (for A = 120).

Nuclear fusion is the process in which two light nuclei combine to form a heavier, more tightly bound nucleus, releasing energy. The proton-proton cycle in the sun fuses four hydrogen nuclei into one helium-4 nucleus, releasing about 26.7 MeV. For fusion to occur, the nuclei must overcome the Coulomb barrier, requiring extremely high temperatures (~10^7–10^9 K), hence the term thermonuclear fusion. Controlled thermonuclear fusion is being actively researched worldwide as a potential source of virtually unlimited clean energy.

Chapter 13 reveals the profound structure underlying atomic nuclei — from the discovery of neutrons and the concept of nuclear binding energy, to the powerful phenomena of radioactivity, fission, and fusion. The binding energy per nucleon curve is the central tool that explains why heavy nuclei split (fission) and why light nuclei merge (fusion) to release energy, both governed by Einstein's mass-energy equivalence E = mc².

For board examinations, this chapter is highly important. Key areas include nuclear radius formula (R = R₀A^(1/3)), mass defect and binding energy calculations, features of the Ebn vs A curve, properties of nuclear forces, types of radioactive decay, Q-value calculations for fission and fusion, and the proton-proton cycle in stars. Numerical problems involving binding energy, energy equivalence (1 u = 931.5 MeV/c²), and fission/fusion energy output are frequently tested.