Class 12 Physics · Chapter 13 NotesNuclei

Every atom has a positively charged nucleus that holds more than 99.9% of the atom's mass, yet its radius is about 10^4 times smaller than the atomic radius. The nucleus contains protons (charge +e, mass 1.00727 u) and neutrons (neutral, mass 1.00866 u), together called nucleons. The atomic number Z equals the number of protons, the neutron number N counts neutrons, and the mass number A = Z + N. A nuclide is written as A over Z X, for example 197 over 79 Au has 79 protons and 118 neutrons. Isotopes have the same Z but different N, isobars have the same A, and isotones have the same N. Atomic masses are measured in atomic mass units, where 1 u is one-twelfth the mass of a carbon-12 atom and equals 1.660539 × 10^-27 kg.

• 1 u = 1/12 mass of one 12C atom = 1.660539 × 10^-27 kg
• Z = number of protons, N = number of neutrons, A = Z + N
• Isotopessame Z, different N. Isobars: same A. Isotones: same N
• Neutron discovered by James Chadwick in 1932 (Nobel Prize 1935)

Rutherford's alpha-particle scattering experiments showed that the positive charge is confined to a very small central region. Electron scattering experiments give the nuclear radius accurately as R = R0 A^(1/3), where R0 = 1.2 × 10^-15 m (1.2 fm). Since volume is proportional to R³, it is proportional to A, so nuclear density is the same for all nuclei, about 2.3 × 10^17 kg m^-3. This is enormously larger than the density of water (10³ kg m^-3) because an atom is mostly empty space. The density of matter in neutron stars is comparable to nuclear density, showing that such matter is compressed almost like one huge nucleus.

• R = R0 A^(1/3), with R0 = 1.2 fm = 1.2 × 10^-15 m
• Nuclear volume ∝ A, so nuclear density is independent of A
• Nuclear density ≈ 2.3 × 10^17 kg m^-3

Einstein's special relativity showed that mass is another form of energy, related by E = mc². One atomic mass unit is equivalent to 931.5 MeV of energy, so even one gram of matter corresponds to 9 × 10^13 J. The mass of a nucleus is always less than the sum of the masses of its separate protons and neutrons; this difference is the mass defect, ΔM = [Z mp + (A − Z) mn] − M. The energy equivalent of the mass defect, Eb = ΔM c², is the binding energy, the energy needed to separate the nucleus into its nucleons. The binding energy per nucleon, Ebn = Eb / A, measures how tightly each nucleon is bound. For middle-mass nuclei (30 < A < 170) Ebn is nearly constant at about 8 MeV, with a maximum of about 8.75 MeV near A = 56, and it is lower for both very light and very heavy nuclei.

• E = mc²; 1 u = 931.5 MeV/c²
• Mass defect ΔM = [Z mp + (A − Z) mn] − M
• Binding energy Eb = ΔM c²; binding energy per nucleon Ebn = Eb / A
• Ebn is about 8 MeV per nucleon for 30 < A < 170

The force that binds nucleons is the strong nuclear force, totally different from the Coulomb force. It is much stronger than the Coulomb repulsion between protons and far stronger than gravitational attraction. It is short-ranged: it falls rapidly to zero beyond a few femtometres, which explains the saturation of nuclear forces and the near-constant binding energy per nucleon in medium and heavy nuclei. The potential energy between two nucleons is minimum at a separation of about 0.8 fm; the force is attractive for larger separations and strongly repulsive for smaller ones. The nuclear force is nearly the same between neutron-neutron, proton-neutron and proton-proton pairs, that is, it does not depend on electric charge. Unlike Coulomb's law or gravitation, it has no simple mathematical form.

• Strong, short-range, charge-independent attractive force
• Attractive beyond about 0.8 fm, repulsive below it
• Saturation explains constant binding energy per nucleon

Radioactivity was discovered accidentally by A. H. Becquerel in 1896 when uranium salts darkened a photographic plate through black paper and silver foil. It is a nuclear phenomenon in which an unstable nucleus undergoes radioactive decay. Three types of decay occur in nature. In alpha decay a helium nucleus (4 over 2 He) is emitted; in beta decay electrons or positrons are emitted; and in gamma decay high-energy photons (hundreds of keV or more) are emitted. The decay constant, half-life and mean life describe how quickly a radioactive sample decays, and activity is measured in becquerel (Bq). Only about 10% of known isotopes are stable; the rest are unstable and decay, or have been produced artificially or observed in astronomical matter.

• Alpha decayemission of a helium nucleus 4 over 2 He
• Beta decayemission of electrons or positrons
• Gamma decayemission of high-energy photons
• Activity R is measured in becquerel (Bq)

Fission is the splitting of a heavy nucleus into two intermediate-mass fragments, usually after absorbing a neutron. For example, a neutron plus 235 over 92 U forms 236 over 92 U, which breaks into fragments such as 144 over 56 Ba and 89 over 36 Kr along with three neutrons, or other pairs like 133 over 51 Sb and 99 over 41 Nb. The fragment nuclei are radioactive and emit beta particles until stable products form. The energy released, the Q value, is about 200 MeV per fissioning nucleus. This can be estimated by comparing binding energies: a nucleus with A = 240 has Ebn about 7.6 MeV, while two A = 120 fragments have Ebn about 8.5 MeV, giving a gain of about 0.9 MeV per nucleon, or roughly 216 MeV in all. The energy first appears as kinetic energy of fragments and neutrons and finally as heat. Nuclear reactors and atom bombs are based on fission.

• Fissionheavy nucleus splits into intermediate-mass fragments
• Q value of uranium fission is about 200 MeV per nucleus
• Energy appears first as kinetic energy, then as heat
• Fission of 1 kg of uranium gives about 10^14 J

Fusion is the joining of two light nuclei to form a heavier, more tightly bound nucleus, releasing energy. Examples include two protons forming a deuteron with a positron and 0.42 MeV, and two deuterons forming either 3 over 2 He plus a neutron (3.27 MeV) or 3 over 1 H plus a proton (4.03 MeV). Because nuclei are positively charged, they must have enough energy to overcome the Coulomb barrier; for two protons this barrier is about 400 keV, corresponding to a temperature of about 3 × 10^9 K. Fusion achieved by heating the fuel to such temperatures is called thermonuclear fusion. In the sun, the proton-proton cycle burns hydrogen into helium, releasing 26.7 MeV when four hydrogen atoms combine to form one helium atom. The sun's core temperature is about 1.5 × 10^7 K, so fusion there involves protons with energies much above average. Controlled thermonuclear fusion aims to produce steady power from plasma at about 10^8 K, but confining such hot plasma is a major challenge.

• Fusionlight nuclei combine to form a heavier nucleus, releasing energy
• Coulomb barrier for two protons is about 400 keV
• Thermonuclear fusion requires temperatures around 10^9 K
• Sun's energy comes from the proton-proton cycle; 4 H → He + 26.7 MeV

Radioactive emissions can damage living tissue because the energetic particles and photons can ionise atoms and break chemical bonds in cells. Alpha particles are heavy and strongly ionising but are stopped by a few centimetres of air or the outer layer of skin; beta particles penetrate further and can harm skin and underlying tissue; gamma rays are highly penetrating and can affect internal organs. The damage depends on the type of radiation, its energy and the dose received. This is why radioactive materials are handled with shielding, distance and time limits, and why exposure is monitored. The chapter also notes that mass-energy interconversion occurs in chemical reactions too, but the mass defects there are about a million times smaller than in nuclear reactions, which is why nuclear processes release far more energy.

• Radiation can ionise atoms and damage cells
• Alphastrongly ionising, low penetration; beta: moderate; gamma: highly penetrating
• Safety relies on shielding, distance, time and dose monitoring
• Nuclear reactions release about a million times more energy than chemical reactions