JEE Physics Nuclei Notes Set 05

Nuclear Size, Composition and Density

The nucleus contains Z protons and N = A - Z neutrons. Isotopes have the same Z, isobars the same A, and isotones the same N.
Nuclear radius \( R=R_0A^{1/3} \) with \( R_0\approx1.2\ \text{fm} \). Volume and mass are both proportional to A, so the density is the same for all nuclei, about \( 2.3\times10^{17}\ \text{kg/m}^3 \). The surface area varies as \( A^{2/3} \).
Atomic mass unit: 1 u = 1/12 of the mass of a carbon-12 atom \( =1.66\times10^{-27}\ \text{kg} \), with energy equivalent 931.5 MeV.
Nuclear forces are the strongest known, short-ranged (about 1 to 2 fm), charge independent and saturated: a nucleon interacts only with its immediate neighbours.

Mass Defect and Binding Energy

The mass of a nucleus is less than the total mass of its separate nucleons. The mass defect is \( \Delta m=Zm_p+(A-Z)m_n-m_{nucleus} \), and the binding energy is \( BE=\Delta m\times931.5\ \text{MeV} \) (mass in u). It is the energy needed to separate the nucleus completely.
The binding energy per nucleon rises quickly for light nuclei, is highest (about 8.8 MeV) near iron-56, then falls slowly for heavy nuclei. Hence energy is released both when very light nuclei fuse and when very heavy nuclei split. A nucleus with larger BE per nucleon is more stable.
Energy released in a reaction \( Q=(\text{total BE of products})-(\text{total BE of reactants}) \) or equivalently \( Q=(\text{mass of reactants}-\text{mass of products})\times931.5\ \text{MeV} \).

Radioactive Decay Law

Each nucleus has a fixed probability per unit time \( \lambda \) of decaying, independent of time, temperature and pressure. The number of undecayed nuclei is \( N=N_0e^{-\lambda t} \), and the activity \( R=\left|\frac{dN}{dt}\right|=\lambda N=R_0e^{-\lambda t} \) (unit: becquerel, 1 Bq = 1 decay per second; 1 curie = \( 3.7\times10^{10} \) Bq).
Half-life \( T_{1/2}=\frac{\ln2}{\lambda}=\frac{0.693}{\lambda} \); mean life \( \tau=\frac1\lambda=1.44\,T_{1/2} \). After n half-lives \( N=\frac{N_0}{2^n} \). After one mean life the fraction left is \( \frac1e \).
If a nuclide decays by two parallel modes, \( \lambda=\lambda_1+\lambda_2 \). If it is produced at a constant rate R while decaying, \( N=\frac{R}{\lambda}(1-e^{-\lambda t}) \), approaching the equilibrium value \( \frac R\lambda \). For a stable end product \( \lambda=0 \).

Alpha, Beta and Gamma Decay

Alpha decay: \( {}^{A}_{Z}X\rightarrow{}^{A-4}_{Z-2}Y+{}^4_2He \). The Q-value is shared inversely to the masses: \( K_\alpha=Q\frac{A-4}{A} \). In a chain, the number of alpha particles is \( \frac{\Delta A}{4} \), after which the number of beta-minus emissions follows from the atomic number.
Beta-minus: a neutron changes to a proton, \( n\rightarrow p+e^-+\bar\nu \); Z increases by 1, A unchanged. Beta-plus: \( p\rightarrow n+e^++\nu \); Z decreases by 1. Electron capture also lowers Z by 1. The beta particle has a continuous energy spectrum up to Q because the neutrino shares the energy. Using atomic masses, \( Q_{\beta^-}=(m_X-m_Y)c^2 \) and \( Q_{\beta^+}=(m_X-m_Y-2m_e)c^2 \).
Gamma decay: an excited nucleus drops to a lower state by emitting a photon; Z and A do not change.

Nuclear Fission and Fusion

Fission: a heavy nucleus such as U-235 absorbs a slow neutron and splits into two medium-mass fragments with 2 or 3 neutrons and about 200 MeV (about 0.9 MeV per nucleon) of energy. The neutrons can cause further fissions in a chain reaction. A moderator (heavy water, graphite) slows neutrons by elastic collisions with light nuclei, control rods (cadmium, boron) absorb neutrons, and the multiplication factor k must equal 1 in a steady reactor.
Fusion: light nuclei combine, for example \( {}^2H+{}^2H\rightarrow{}^3H+p \), \( {}^2H+{}^3H\rightarrow{}^4He+n \) (17.6 MeV). It needs a temperature of about \( 10^7 \) to \( 10^8 \) K to overcome Coulomb repulsion and is the source of stellar energy.

Worked Examples

Worked Example 1. Find the binding energy per nucleon of He-4 if its mass is 4.0026 u (atomic) and the atomic masses of hydrogen and the neutron are 1.0078 u and 1.0087 u.
Answer:
Mass defect \( =2(1.0078)+2(1.0087)-4.0026=0.0304\ \text{u} \).
BE \( =0.0304\times931.5=28.3\ \text{MeV} \), so per nucleon \( \approx7.1\ \text{MeV} \).

Worked Example 2. The half-life of a radioactive sample is 20 minutes. What fraction remains after 1 hour, and what is its activity relative to the initial activity?
Answer:
1 hour = 3 half-lives, so \( \frac{N}{N_0}=\frac{1}{2^3}=\frac18 \).
Activity is proportional to N, so the activity is also one-eighth of the initial value.

Worked Example 3. U-238 decays to Pb-206. How many alpha and beta-minus particles are emitted?
Answer:
Alpha: \( \frac{238-206}{4}=8 \); these lower Z from 92 to 76. To reach Z = 82, six beta-minus particles are emitted.

Worked Example 4. A radioactive source has activity 4000 Bq. After 10 days it is 1000 Bq. Find the half-life.
Answer:
The activity fell to one quarter, i.e. two half-lives, so \( T_{1/2}=\frac{10}{2}=5 \) days.