Nuclei

In this chapter: atomic mass unit, composition of the nucleus, isotopes, isobars and isotones, size and density of nuclei, mass-energy equivalence, mass defect and binding energy, binding energy per nucleon curve, nuclear force, nuclear fission, nuclear fusion and energy in stars, with a short note on radioactive decay.

Atomic masses and composition of the nucleus

Atomic masses are measured in the atomic mass unit, defined as one-twelfth of the mass of a carbon-12 atom:

Masses are measured accurately with a mass spectrometer. Many elements have atomic masses that are not whole numbers because they are mixtures of isotopes; chlorine, for example, is about 75.4% Cl-35 and 24.6% Cl-37, giving an average of 35.47 u.

A nucleus contains Z protons and N neutrons (together called nucleons). The mass number is A = Z + N, and the nucleus is written ^A_ZX. The neutron was discovered by Chadwick in 1932; it is neutral, slightly heavier than the proton, and stable inside a nucleus though a free neutron decays with a mean life of about 1000 s.

Size of the nucleus

Scattering experiments with fast electrons and alpha particles show that the nuclear radius grows with mass number as

So the volume (∝ R³) is proportional to A, which means the density of nuclear matter is the same for all nuclei, about 2.3 × 10¹⁷ kg m⁻³. That is roughly 10¹⁴ times the density of ordinary matter; a neutron star is an example of matter at this density.

Worked example: Find the ratio of the nuclear radii of ²⁷Al and ¹²⁵Te.
Solution: R_Al/R_Te = (27/125)^1/3 = 3/5. With R₀ = 1.2 fm, R_Al = 3.6 fm and R_Te = 6.0 fm.

Mass-energy and nuclear binding energy

Einstein's relation E = mc² says mass is a form of energy. The energy equivalent of 1 u is

The mass of a nucleus is always less than the total mass of its separate protons and neutrons. The difference is the mass defect:

E_b is the energy released when the nucleus is formed from its nucleons, and equally the energy needed to pull it apart into separate nucleons. Tables usually give atomic masses, which include the electrons; in that case use the mass of a hydrogen atom m_H in place of m_p, so the electron masses cancel.

Worked example: The atomic mass of ¹⁶O is 15.99493 u. Find its binding energy and binding energy per nucleon.
Solution: Mass of 8 hydrogen atoms and 8 neutrons = 8 × 1.00783 + 8 × 1.00866 = 8.06264 + 8.06928 = 16.13192 u. ΔM = 16.13192 − 15.99493 = 0.13699 u. E_b = 0.13699 × 931.5 ≈ 127.6 MeV, and per nucleon 127.6/16 ≈ 7.98 MeV.

Binding energy per nucleon

The binding energy per nucleon, E_bn = E_b/A, measures how tightly a nucleus is held together. Its variation with A shows these features:

• For middle-mass nuclei (30 < A < 170) it is nearly constant at about 8 MeV, with a maximum of about 8.75 MeV near A = 56 (iron).
• It is lower for light nuclei (A < 30) and for heavy nuclei (A > 170). For uranium it is about 7.6 MeV.
• The nearly constant value in the middle shows that nuclear force is short-ranged and saturates: each nucleon interacts only with its close neighbours, so adding more nucleons does not raise the energy per nucleon.

Nuclear force

• It is the strongest force in nature, much stronger than the Coulomb repulsion between protons; that is how it holds a nucleus together.
• It is short-range, acting only over a few femtometres. Beyond that it becomes negligible.
• It is attractive for separations greater than about 0.8 fm and strongly repulsive for smaller separations, which keeps nucleons from crowding closer. The potential energy is minimum at about 0.8 fm.
• It is nearly the same between neutron-neutron, proton-neutron and proton-proton pairs: it does not depend on electric charge.
• Unlike gravitation and the Coulomb force, it has no simple mathematical form.

Nuclear energy

Chemical reactions involve energies of a few eV per atom; nuclear reactions involve MeV per nucleus, roughly a million times more. Energy is released in any nuclear change that increases the total binding energy.

Fission

When a slow neutron is absorbed by ²³⁵U, the nucleus becomes unstable and splits into two medium-mass fragments, releasing more neutrons, for example

The binding energy per nucleon rises from about 7.6 MeV for uranium to about 8.5 MeV for the fragments, so roughly 0.9 × 236 ≈ 200 MeV is released per fission. Most of it appears as kinetic energy of the fragments, which heats the surroundings. The extra neutrons can cause further fissions, giving a chain reaction; in a nuclear reactor this chain is controlled to produce steady power.

Fusion and energy in stars

When two light nuclei combine to form a heavier one, the binding energy per nucleon increases and energy is released:

Both nuclei are positively charged, so they must approach each other against the Coulomb barrier. This needs very high temperatures, which is why the process is called thermonuclear fusion. In the core of the sun (about 1.5 × 10⁷ K), the proton-proton cycle converts hydrogen into helium; the net result is

Controlled fusion on earth, which would give a nearly limitless source of power, has not yet been achieved on a practical scale.

A note on radioactive decay

Radioactivity (alpha, beta and gamma decay and the decay law) is not part of the rationalised NCERT chapter, but it is useful background and still appears in many question banks. Check the current syllabus before spending time on it. The decay law is N = N₀e^−λt, with half-life T_1/2 = 0.693/λ, mean life τ = 1/λ and activity R = λN (SI unit becquerel, 1 Bq = 1 decay per second). After n half-lives, a fraction (1/2)ⁿ of the nuclei remains.

Common mistakes: (1) Mixing atomic and nuclear masses in the mass defect; with atomic masses, use m_H instead of m_p. (2) Thinking the most tightly bound nucleus is the one with the largest total binding energy; it is the one with the largest binding energy per nucleon. (3) Writing R ∝ A instead of R ∝ A^1/3. (4) Saying nuclear density increases with A; it is the same for all nuclei. (5) Saying fusion occurs because nuclei attract at any distance; they must first overcome Coulomb repulsion.

JEE and NEET focus

• Nuclear radius and the constancy of nuclear density.
• Mass defect, binding energy and binding energy per nucleon from given masses.
• Shape of the binding energy curve and the reason fission and fusion release energy.
• Properties of nuclear force.
• Energy released (Q value) in fission and fusion reactions from masses or binding energies.

Practice questions