JEE Physics Nuclei Notes Set 02

Structure of the Nucleus

The nucleus is the tiny, dense core of the atom. It contains protons and neutrons, together called nucleons. A nuclide is written \( {}^{A}_{Z}\mathrm{X} \), where Z is the atomic number (number of protons) and A is the mass number (protons plus neutrons), so the number of neutrons is N = A − Z. Nuclides with the same Z are isotopes, with the same A are isobars and with the same N are isotones.

Masses are measured in the atomic mass unit u, defined as 1/12 of the mass of a carbon-12 atom: 1 u = 1.66 × 10⁻²⁷ kg, which is equivalent to 931 MeV/c². A proton weighs about 1.00727 u and a neutron about 1.00866 u.

Size and Density

Experiments show that the radius of a nucleus is R = R₀A^1/3 with R₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m). Since the volume is proportional to A and the mass is also proportional to A, the density is the same for every nucleus, about 2.3 × 10¹⁷ kg/m³. Radii of two nuclei compare as R₁/R₂ = (A₁/A₂)^1/3. The atom as a whole is about 10⁵ times larger in radius, so almost all of its volume is empty space.

Example. A nucleus of mass number 64 has a radius (64/8)^1/3 = 2 times that of a nucleus of mass number 8. For A = 27, R = 1.2 × 27^1/3 = 3.6 fm.

Nuclear Force

Three forces act in a nucleus: gravity (very weak), the electrostatic repulsion between protons, and the strong nuclear force which holds the nucleons together. The nuclear force has these main features.

1. It is the strongest of the forces and acts between all nucleons: proton–proton, neutron–neutron and proton–neutron.
2. It has a very short range, about 2 to 3 fm, and is almost zero beyond that.
3. It is charge independent and spin dependent.
4. It is not a central force.
5. It is attractive for separations larger than about 0.8 fm and repulsive for smaller separations.
6. It does not obey an inverse square law and is saturated: a nucleon interacts mainly with its near neighbours.

Because the Coulomb repulsion has long range while the nuclear force has short range, very heavy nuclei become unstable as the number of protons grows.

Mass Defect and Binding Energy

The mass of a nucleus is always less than the total mass of its separate nucleons. The difference is the mass defect: Δm = [Z m_p + (A − Z) m_n] − M. When atomic masses are used, the mass of Z hydrogen atoms replaces Z protons, which takes care of the electrons: Δm = [Z m_H + (A − Z) m_n] − M_atom.

The binding energy is the energy needed to separate the nucleus into free nucleons, E_b = Δm c², or E_b = Δm × 931 MeV when Δm is in u. The binding energy per nucleon, E_b/A, measures stability: the larger it is, the more stable the nucleus.

Example. For the deuteron: m_H = 1.007825 u, m_n = 1.008665 u and M(²H) = 2.014102 u. Then Δm = 1.007825 + 1.008665 − 2.014102 = 0.002388 u, so E_b = 0.002388 × 931 = 2.22 MeV, or 1.11 MeV per nucleon.

Binding Energy Curve

When E_b/A is plotted against A, it rises steeply for the lightest nuclei, shows peaks at nuclei such as He-4 and O-16, reaches a broad maximum of about 8.8 MeV per nucleon near A = 56 (iron) and then falls slowly to about 7.6 MeV per nucleon for uranium. Three conclusions follow.

1. Nuclei near iron are the most stable.
2. Fusion of two light nuclei gives a more tightly bound nucleus and releases energy.
3. Fission of a heavy nucleus into two medium-mass nuclei also releases energy.

In both cases energy released = (total binding energy of products) − (total binding energy of reactants). The Q value of a reaction is Q = (mass of reactants − mass of products)c²; Q > 0 means exothermic and Q < 0 means endothermic.

Example. A nucleus of A = 240 with 7.6 MeV per nucleon splits into two nuclei of A = 120 each having 8.5 MeV per nucleon. Energy released = 240 × 8.5 − 240 × 7.6 = 216 MeV.

Nuclear Fission

Fission is the splitting of a heavy nucleus, usually after absorbing a neutron, into two lighter nuclei with the release of two or three neutrons and about 200 MeV of energy. A typical reaction is \( {}^{235}_{92}\mathrm{U} + {}^{1}_{0}n \rightarrow {}^{144}_{56}\mathrm{Ba} + {}^{89}_{36}\mathrm{Kr} + 3\,{}^{1}_{0}n + Q \). About 93% of the energy appears as kinetic energy of the fragments, and the mass converted is about 0.1% of the mass of the nucleus. Natural uranium has 0.7% U-235 and 99.3% U-238; U-235 is fissioned by slow (thermal) neutrons, while U-238 only absorbs fast neutrons.

The neutrons released can cause further fissions, producing a chain reaction. The multiplication (reproduction) factor is k = (rate of production of neutrons)/(rate of loss of neutrons). If k = 1 the reaction is steady (critical, as in a reactor), if k > 1 it grows rapidly (super-critical, as in an atom bomb), and if k < 1 it dies out (sub-critical).

Example. A 500 MW reactor with 200 MeV per fission needs 500 × 10⁶/(200 × 1.6 × 10⁻¹³) = 1.6 × 10¹⁹ fissions per second.

Nuclear Reactor

A reactor sustains a controlled chain reaction. Its parts and their purposes are listed below.

1. Fuel: enriched uranium (U-235) or plutonium-239.
2. Moderator: heavy water or graphite, light materials which slow neutrons down by elastic collisions without absorbing them.
3. Control rods: boron or cadmium, which absorb neutrons and are moved in or out to keep k = 1.
4. Coolant: water or a liquid metal, which carries heat away to produce steam.
5. Shielding: concrete and lead to absorb radiation.

In a fast breeder reactor, U-238 captures a neutron to become U-239, which changes by two β⁻ decays to Pu-239, a new fissile fuel. Such a reactor makes more fuel than it uses.

Nuclear Fusion

Fusion is the joining of two light nuclei into a heavier one with release of energy, for example four hydrogen nuclei forming one helium nucleus with about 26 MeV released. The product has a larger binding energy per nucleon than the reactants. Fusion needs temperatures of about 10⁷ K and high density so that the nuclei can overcome the Coulomb repulsion; these conditions exist in the Sun and stars, and in a hydrogen bomb. Per fission about 200 MeV is released but per nucleon only about 0.85 MeV, whereas fusion gives about 6.5 MeV per nucleon, so fusion releases more energy per unit mass.

Pair Production and Annihilation

A γ-photon with energy of at least 1.02 MeV (twice the electron rest energy of 0.51 MeV), passing near a nucleus, can create an electron–positron pair. Any extra energy appears as kinetic energy of the pair, each getting (E − 1.02)/2 MeV. The reverse process, annihilation, occurs when an electron and a positron meet and vanish into two γ-photons of 0.51 MeV each, in opposite directions.

Key Facts to Remember

1. 1 u = 931 MeV/c² = 1.66 × 10⁻²⁷ kg and 1 MeV = 1.6 × 10⁻¹³ J.
2. Energy equivalent of mass: E = mc².
3. Nuclear density is about 10¹⁷ kg/m³ and is independent of A.
4. Energy per fission is about 200 MeV; fissions per second = power/(energy per fission).
5. A free neutron is unstable (n → p + e⁻ + ν̄), a free proton is stable.
6. Maximum binding energy per nucleon is about 8.8 MeV at Fe-56.