JEE Physics Nuclei Notes Set 04

Nuclear Size, Mass and Binding Energy

1. Radius \( R=R_0A^{1/3} \) with \( R_0\approx1.2 \) fm. Volume is proportional to A, so the density is nearly the same for all nuclei.
2. Mass defect \( \Delta m=Zm_p+(A-Z)m_n-M \) is positive for every stable nucleus. Binding energy is \( \Delta m\,c^2 \), and 1 u corresponds to 931 MeV.
3. Binding energy per nucleon rises with A to a maximum near iron (about 8.7 MeV at A = 56) and then falls slowly. This is why fusion of light nuclei and fission of heavy nuclei both release energy.
4. Energy released in a nuclear reaction is the difference of the total binding energies: for two deuterons forming He-4, \( 28-2\times2.2=23.6 \) MeV.

Radioactive Decay

1. After n half-lives the fraction left is \( \tfrac1{2^n} \). To reduce to 1 percent takes \( \log_2 100 \approx 6.64 \) half-lives.
2. Alpha decay lowers A by 4 and Z by 2. Beta-minus decay raises Z by 1 with A unchanged. Gamma emission changes neither A nor Z.
3. For a chain from mass number \( A_1 \) to \( A_2 \): number of alpha particles \( =\dfrac{A_1-A_2}{4} \), then beta particles are found from the change in Z.
4. In alpha decay of a nucleus at rest, the momenta are equal and opposite, so the kinetic energies are in the ratio of the inverse masses: the alpha particle gets \( \dfrac{A-4}{A} \) of the energy released.
5. The continuous energy spectrum of beta particles (zero up to a maximum \( E_0 \)) led to the neutrino hypothesis.

Fission, Fusion and Reactors

1. Fission fragments are neutron rich and undergo beta-minus decay. Neutrons released can start a chain reaction.
2. In a reactor, a moderator (heavy water, graphite) slows neutrons down, control rods (cadmium, boron) absorb neutrons to keep the chain reaction steady, and a coolant removes the heat.
3. Fusion needs extremely high temperatures so that nuclei overcome their electrical repulsion, as in the Sun.