JEE Physics Nuclei Notes Set 01

Atom and Nucleus

An atom has a tiny, dense, positively charged nucleus at its centre with electrons in shells around it. The nucleus (size about 10^-15 m to 10^-14 m) holds protons and neutrons, together called nucleons. The atomic number Z is the number of protons (equal to the number of electrons in a neutral atom) and the mass number A is the total number of nucleons, so the neutron number is A - Z. An element X is written as \( {}^{A}_{Z}X \). Shell n can hold at most 2n² electrons. An electron has charge \( -1.6\times10^{-19} \) C and mass about 1/1840 of the proton.

Isotopes, Isobars and Isotones

Isotopes have the same Z but different A (different numbers of neutrons) and therefore the same chemical properties; for example protium, deuterium and tritium of hydrogen, with 0, 1 and 2 neutrons. Some isotopes are stable and others are radioactive.
Isobars have the same A but different Z, for example \( {}^{23}_{11}Na \) and \( {}^{23}_{12}Mg \). Isotones have the same number of neutrons (A - Z) but different Z and A; \( {}^{23}_{11}Na \) with 12 neutrons and \( {}^{24}_{12}Mg \) with 12 neutrons form an isotone pair.

Radioactivity

Radioactivity is the spontaneous disintegration of unstable nuclei with emission of radiation. Nearly all elements with Z above 82 are radioactive (uranium, radium, polonium, thorium). The process is not affected by chemical changes, temperature or pressure, which shows that it is a nuclear phenomenon and not due to orbital electrons. Three types of radiation are emitted: alpha, beta and gamma. In a magnetic or electric field, alpha particles bend one way, beta particles the opposite way and by a larger angle, while gamma rays go undeflected.

Properties of Alpha, Beta and Gamma Radiation

Alpha particle: a helium nucleus \( {}^{4}_{2}He \) (two protons and two neutrons), charge +2e, speed about \( 10^{7} \) m/s. It has the highest ionising power (about 100 times that of beta and \( 10^{4} \) times that of gamma) and the lowest penetrating power, and is stopped by a sheet of paper. It is deflected by electric and magnetic fields and affects photographic plates.
Beta particle: a fast electron \( {}^{0}_{-1}e \) emitted from the nucleus (unlike cathode rays, which come from the orbital electrons of atoms), with speed up to about \( 10^{8} \) m/s. Its ionising power is about 1/100 that of alpha and its penetrating power is greater, about 100 times that of alpha. It is deflected strongly, being lighter than alpha. Beta particles stopped by a heavy metal such as tungsten produce X-rays.
Gamma radiation: electromagnetic waves of very short wavelength, speed equal to that of light, no charge, so no deflection by fields. Ionising power is the lowest (about \( 10^{-4} \) that of alpha) and penetrating power the highest (about \( 10^{4} \) times alpha and \( 10^{2} \) times beta). Gamma rays are diffracted by crystals and are used to treat cancer.

Changes in the Nucleus

Alpha decay: \( {}^{A}_{Z}X \rightarrow {}^{A-4}_{Z-2}Y + {}^{4}_{2}He \). The mass number falls by 4 and the atomic number by 2.
Beta decay: a neutron changes to a proton inside the nucleus, so \( {}^{A}_{Z}X \rightarrow {}^{\;\;A}_{Z+1}Y + {}^{0}_{-1}e \). The mass number is unchanged and the atomic number rises by 1.
Gamma emission: an excited nucleus \( {}^{A}_{Z}X^{*} \) drops to a lower energy state and emits a photon, \( {}^{A}_{Z}X^{*} \rightarrow {}^{A}_{Z}X + \gamma \), with no change in A or Z.

Worked example 1. A nucleus \( {}^{238}_{92}U \) emits an alpha particle. Find the daughter nucleus.
Answer:
Mass number: 238 - 4 = 234. Atomic number: 92 - 2 = 90.
The daughter is \( {}^{234}_{90}Th \).

Worked example 2. \( {}^{234}_{90}Th \) undergoes beta decay. Find the daughter nucleus.
Answer:
Mass number stays 234. Atomic number becomes 90 + 1 = 91.
The daughter is \( {}^{234}_{91}Pa \).

Uses of Radioisotopes

Medical: cobalt-60 gamma rays for cancer therapy; tracers such as radio-sodium, radio-iron and radio-iodine for diagnosis; gamma sterilisation of dressings and instruments. Scientific: alpha particles as projectiles in nuclear reactions and for estimating nuclear size; tracers in agriculture for studying uptake of fertilisers; carbon-14 dating of dead plants and fossils. Industrial: fuel for reactors, removing static charge, luminescent signs, and controlling the thickness of paper, plastic and metal sheets using the penetration of beta radiation.

Harmful Effects and Safety

Radiation can cause short-term effects (nausea, diarrhoea, loss of hair), long-term effects (leukaemia, cancer) and genetic damage. Workers use lead-lined aprons and gloves and handle sources with long lead tongs, store them in thick lead containers with a narrow opening and keep exposure below safe limits. Power reactors are shielded by lead and steel, housed in strong concrete buildings and have back-up cooling systems. Nuclear waste is stored in thick casks and buried deep underground in places far from populated areas.

Mass-Energy Equivalence

In nuclear changes, the total mass of the products is slightly less than the mass of the reactants. This mass loss \( \Delta m \) appears as energy: \( E = \Delta m\,c^2 \) (Einstein). One kilogram of mass is equivalent to \( 9\times10^{16} \) J and 1 atomic mass unit is equivalent to 931 MeV.

Nuclear Fission

Fission is the splitting of a heavy nucleus into two lighter nuclei of comparable size when it is hit by a slow neutron, releasing a large amount of energy. It was discovered by Otto Hahn and Fritz Strassmann (1938). A typical reaction: \( {}^{235}_{92}U + {}^{1}_{0}n \rightarrow {}^{144}_{56}Ba + {}^{89}_{36}Kr + 3\,{}^{1}_{0}n + \text{energy} \).
Energy estimate with masses 235 U = 234.99 u, n = 1.01 u, 144 Ba = 143.87 u, 89 Kr = 88.90 u: mass before = 236.00 u, mass after = 143.87 + 88.90 + 3.03 = 235.80 u, so \( \Delta m = 0.20 \) u and \( E = 0.20\times931 \approx 190 \) MeV (about 200 MeV with more precise masses). Most of this is the kinetic energy of the fragments.
Chain reaction: the neutrons released cause further fissions. If uncontrolled, the energy is released explosively (the principle of the atom bomb). In a reactor, a moderator such as graphite or heavy water slows neutrons and control rods absorb them, so that the reaction proceeds at a steady rate and the energy generates electric power.

Nuclear Fusion

Fusion is the joining of two light nuclei to form a heavier nucleus with release of energy, because the product has less mass than the sum of the parts. Example: \( {}^{2}_{1}H + {}^{2}_{1}H \rightarrow {}^{3}_{2}He + {}^{1}_{0}n + 3.3 \) MeV, followed by \( {}^{3}_{2}He + {}^{2}_{1}H \rightarrow {}^{4}_{2}He + {}^{1}_{1}H + 18.3 \) MeV. Fusion needs a very high temperature (about \( 10^{7} \) K) and pressure to overcome the electrostatic repulsion between nuclei; it powers the sun and the hydrogen bomb.

Fission and Fusion Compared