Nuclei notes: CBSE Class 12 Physics

1. Composition and size of nuclei

Nucleus: $Z$ protons and $N = A - Z$ neutrons (nucleons); notation ${}^{A}_{Z}\text{X}$ .
Isotopes: same Z, different A. Isobars: same A, different Z. Isotones: same N.
Masses in unified atomic mass units: 1 u = 1/12 of the mass of a ¹²C atom = $1.6605 \times 10^{-27}$ kg; $m_p = 1.007276$ u, $m_n = 1.008665$ u, $m_\text{H} = 1.007825$ u.
Radius: $R = R_0A^{1/3}$ , $R_0 \approx 1.2$ fm. Volume ∝ A, so nuclear density is the same for all nuclei, about $2.3 \times 10^{17}$ kg m⁻³.

Nucleon: A proton or a neutron.
Isotopes: Nuclides of the same element (same Z) with different mass numbers.

Holds nucleons together against the Coulomb repulsion of protons; much stronger than the electrostatic force at nuclear distances.
Short range: strongly attractive around 1 fm, negligible beyond a few fm; repulsive below about 0.8 fm (so nuclei do not collapse).
Charge-independent: p–p, n–n and n–p forces are about equal.
Saturation: each nucleon interacts only with its nearest neighbours.

Attractive between about 0.8 fm and a few fm, repulsive at shorter range; negligible beyond a few fm (short range).

Nuclear force: The strong, short-range, charge-independent attraction between nucleons.

$E = mc^2$ ; 1 u ≈ 931.5 MeV; 1 MeV = $1.6 \times 10^{-13}$ J.
Mass defect: $\Delta M = Zm_\text{H} + (A - Z)m_n - M_\text{atom}$ (with atomic masses).
Binding energy: $E_b = \Delta M c^2 = \Delta M \times 931.5$ MeV/u: the energy to separate the nucleus into its nucleons.
Binding energy per nucleon $E_{bn} = E_b/A$ : about 7.07 MeV for ⁴He, 7.98 MeV for ¹⁶O, a peak ≈ 8.8 MeV near A = 56, about 7.6 MeV for ²³⁵U and ²³⁸U.
Nearly constant $E_{bn}$ for 30 < A < 170 shows the short range (saturation) of the nuclear force; the fall for heavy nuclei is due to Coulomb repulsion.

Fusion of light nuclei and fission of heavy nuclei both move towards the peak, so both release energy.

Mass defect: The difference between the total mass of the separate nucleons and the mass of the nucleus.
Binding energy: Energy equivalent of the mass defect; energy needed to break the nucleus into free nucleons.

Fission: a heavy nucleus splits into two medium nuclei, e.g. $\text{n} + {}^{235}\text{U} \to {}^{141}\text{Ba} + {}^{92}\text{Kr} + 3\text{n}$ ; about 200 MeV per fission (≈ 0.85 MeV per nucleon). The neutrons can sustain a chain reaction: uncontrolled in a bomb, controlled in a reactor (moderator slows neutrons; control rods absorb them; coolant removes heat).
Fusion: light nuclei join, e.g. $\,{}^{2}\text{H} + {}^{3}\text{H} \to {}^{4}\text{He} + \text{n} + 17.6$ MeV (≈ 3.5 MeV per nucleon). Needs $10^7$ – $10^8$ K to overcome Coulomb repulsion (thermonuclear). Powers the Sun: net $4\,{}^{1}\text{H} \to {}^{4}\text{He}$ + 26.7 MeV.
Why both release energy: the products lie nearer the peak of the BE-per-nucleon curve, so they are more tightly bound; the gain in total binding energy is released.
$Q$ = (mass before − mass after) × 931.5 MeV/u = (total BE after − total BE before).

Energy released
$Q = \left(\sum m_\text{before} - \sum m_\text{after}\right) \times 931.5\ \text{MeV/u}$

Fission: Splitting of a heavy nucleus into two lighter nuclei with release of energy.
Fusion: Joining of light nuclei into a heavier nucleus with release of energy.
Chain reaction: A self-sustaining series of fissions caused by the neutrons from earlier fissions.

Use $m_\text{H}$ with atomic masses; keep six decimal places until the end.
1 u = 931.5 MeV; 1 MeV = $1.6 \times 10^{-13}$ J.
Sketch the BE curve: peak ≈ 8.8 MeV near A = 56; explain fission and fusion from it.
Balance A and Z in every nuclear equation.