JEE Physics Nuclei Notes Set 06

Nuclear Size, Mass and Density

A nucleus of mass number A has radius \( R=R_0A^{1/3} \) with \( R_0\approx1.2\ \text{fm} \). Its volume is proportional to A, and so is its mass, so the density of nuclear matter is the same for all nuclei, about \( 10^{17}\ \text{kg/m}^3 \).
Mass defect \( \Delta m=Zm_p+(A-Z)m_n-m_{nucleus} \). Binding energy \( =\Delta m\times931.5 \) MeV, with masses in u. The total binding energy is the binding energy per nucleon times A.

Radioactive Decay

Decay law: \( N=N_0e^{-\lambda t} \). Half-life \( T_{1/2}=\frac{0.693}{\lambda} \), so after n half-lives the fraction left is \( \left(\frac12\right)^n \). Mean life is \( \tau=\frac1\lambda \).
If a nucleus can decay in two ways at once, the decay constants add: \( \lambda=\lambda_1+\lambda_2 \), hence \( \frac1T=\frac1{T_1}+\frac1{T_2} \).
Alpha decay lowers A by 4 and Z by 2. Beta-minus raises Z by 1, beta-plus lowers Z by 1, with A unchanged. Gamma emission changes neither A nor Z. A free neutron can decay into a proton, an electron and an antineutrino because it is heavier than a proton.

Nuclear Reactions and Fission

In fission a heavy nucleus splits into two lighter fragments and releases about 200 MeV per nucleus. Momentum is conserved, so when a nucleus at rest splits, the lighter fragment moves faster.
To find the energy from a sample, count the nuclei using \( N=\frac{m}{M}N_A \) and multiply by the energy per fission.