JEE Physics Nuclei Notes Set 08

Nuclear Structure and Size

A nucleus of atomic number Z and mass number A contains Z protons and \( N=A-Z \) neutrons. Nuclides with equal Z are isotopes, with equal A are isobars and with equal N are isotones. The radius is \( R=R_0A^{1/3} \) with \( R_0\approx1.2 \) fermi (1 fermi = \( 10^{-15} \) m), so volume is proportional to A and the density, about \( 2.3\times10^{17} \) kg/m\( ^3 \), is the same for all nuclei. For example, a nucleus of mass number 125 has a radius \( \frac53 \) times that of mass number 27. The unified atomic mass unit satisfies 1 u = 931.5 MeV.

Mass Defect and Binding Energy

The mass of a nucleus is less than the sum of the masses of its nucleons by the mass defect \( \Delta m \), and the binding energy is \( B=\Delta m\,c^2 \). The binding energy per nucleon rises quickly for light nuclei, is about 8 MeV for the middle of the table and peaks near iron (mass number 56), then falls slowly for heavy nuclei. Fusion of light nuclei and fission of heavy nuclei both move towards the peak and release energy. The energy released in a reaction equals the binding energy of the products minus that of the reactants; for example, \( {}^7\text{Li}+p\to2\,{}^4\text{He} \) with 5.60 and 7.06 MeV per nucleon gives \( 56.48-39.2=17.3 \) MeV.

Radioactive Decay

The number of undecayed nuclei is \( N=N_0e^{-\lambda t} \), the activity is \( A=\lambda N \), the half-life is \( T_{1/2}=\frac{\ln2}{\lambda}=0.693\tau \) and the mean life is \( \tau=\frac1\lambda \). After n half-lives the fraction left is \( \left(\frac12\right)^n \). If a sample decays by two independent routes, the decay constants add. Decay of an individual nucleus is random, and the half-life is unaffected by temperature, pressure or chemical state. The age of a rock is found from the ratio of parent to daughter nuclei, with the original parent number equal to parent plus daughter.
Alpha emission lowers A by 4 and Z by 2, beta-minus emission raises Z by 1 (a neutron changes into a proton, electron and antineutrino), beta-plus lowers Z by 1, and gamma emission changes only the energy of the nucleus. To count alpha and beta emissions, divide the change in A by 4 first, then restore Z with beta emissions. Alpha spectra are discrete, whereas beta spectra are continuous.

Fission and Fusion

A heavy nucleus such as U-235, after absorbing a slow neutron, splits into two medium nuclei plus two or three neutrons and releases about 200 MeV; a chain reaction occurs above a critical mass. In a reactor a moderator (heavy water or graphite) slows the neutrons to thermal energies, control rods absorb neutrons to regulate the rate, and a coolant removes heat. Fusion of light nuclei needs temperatures of about \( 10^7 \) to \( 10^9 \) K to overcome Coulomb repulsion, and powers the Sun through the proton-proton chain (26.7 MeV per helium nucleus formed). The Lawson criterion requires that the product of particle density and confinement time exceeds a minimum number.

Worked Examples

A sample falls to 1/16 of its activity in 2 hours: that is 4 half-lives, so the half-life is 30 minutes.
The uranium-238 to lead-206 series involves \( \frac{238-206}{4}=8 \) alpha particles, which lower Z by 16 to 76, so 6 beta particles restore Z to 82.
An alpha particle emitted from a nucleus of mass number 220 at rest with Q = 5.5 MeV carries \( 5.5\times\frac{216}{220}=5.4 \) MeV.