JEE Physics Nuclei Notes Set 09

The Nucleus

A nucleus is made of protons and neutrons, together called nucleons. A nucleus is written \( {}^{A}_{Z}\mathrm{X} \): Z is the number of protons, A is the mass number (number of nucleons) and the number of neutrons is \( N=A-Z \).
The atomic mass unit is one twelfth of the mass of a carbon-12 atom: \( 1\ \mathrm{u}=1.66\times10^{-27}\ \mathrm{kg}\) which corresponds to 931.5 MeV (that is, \( 1\ \mathrm{u}\,c^2 = 931.5\ \mathrm{MeV} \)). Proton and neutron masses are each close to 1 u.
Isotopes have the same Z and different A; isobars have the same A and different Z; isotones have the same number of neutrons.

Size and Density of the Nucleus

Nuclear radius grows with the cube root of the mass number: \( R=R_0A^{1/3} \) with \( R_0\approx1.1\ \text{fm}\ (1\ \mathrm{fm}=10^{-15}\ \mathrm m) \).
Because volume is proportional to A and mass is proportional to A, the density does not depend on A: \( \rho=\frac{3m_p}{4\pi R_0^3}\approx 3\times10^{17}\ \mathrm{kg/m^3} \). All nuclei have nearly the same enormous density.
Mean atomic mass of an element with several isotopes is the abundance-weighted average of the isotope masses.

Mass Defect and Binding Energy

A nucleus is lighter than the sum of the masses of its separate nucleons. The difference is the mass defect \( \Delta m=Zm_p+(A-Z)m_n-m_{nucleus} \), and the energy equivalent \( B=\Delta m\,c^2 \) is the binding energy, the work needed to pull the nucleus apart into free nucleons.
With atomic masses use hydrogen-atom mass in place of the proton mass: \( \Delta m=Zm_H+(A-Z)m_n-M_{atom} \); the electron masses cancel.
Binding energy per nucleon \( B/A \) rises steeply for light nuclei, reaches a broad maximum of about 8.8 MeV near iron (A about 56), and then drops slowly to about 7.6 MeV for the heaviest nuclei. Nuclei near the peak are the most stable.
Heavy nuclei can release energy by splitting into medium nuclei (fission) and light nuclei can release energy by joining into heavier ones (fusion). Energy released in a reaction equals the gain in total binding energy, or the loss in total rest mass times \( c^2 \).

Radioactivity and the Three Radiations

Radioactivity is the spontaneous emission of radiation by unstable nuclei. It is a nuclear property and does not depend on temperature, pressure or chemical state.
Alpha particle: a helium nucleus (2 protons, 2 neutrons), charge +2e, emitted mostly by nuclei with A above about 210. Beta-minus particle: an electron, charge −e. Beta-plus particle: a positron, charge +e, the antiparticle of the electron. Gamma ray: a high energy photon, no charge, no rest mass.
Gamma rays are weakened by matter as \( I=I_0e^{-\mu x} \), where \( \mu \) is the absorption coefficient of the slab.

Alpha Decay

\( {}^{A}_{Z}\mathrm{X}\to{}^{A-4}_{Z-2}\mathrm{Y}+{}^{4}_{2}\mathrm{He}+Q \), with \( Q=[M_X-M_Y-M_{He}]c^2 \) using atomic masses.
The parent is at rest, so the two products have equal and opposite momenta. Kinetic energy is then shared inversely with mass: \( K_\alpha=\frac{A-4}{A}Q \) and \( K_Y=\frac{4}{A}Q \). Hence every alpha particle from a given decay has the same energy and the same radius in a magnetic field, \( r=\frac{mv}{qB}=\frac{\sqrt{2Km}}{2eB} \).

Beta Decay, Electron Capture and Gamma Decay

Beta-minus: a neutron turns into a proton, \( n\to p+e^-+\bar\nu \); \( Q=[M_X-M_Y]c^2 \) with atomic masses.
Beta-plus: a proton turns into a neutron, \( p\to n+e^++\nu \); \( Q=[M_X-M_Y-2m_e]c^2 \) with atomic masses, so it needs a mass difference of at least \( 2m_e c^2=1.022\ \mathrm{MeV} \).
The beta particles have a continuous energy spectrum from zero up to a maximum equal to the available Q-value. This is why a third particle, the neutrino or antineutrino, must also be emitted to share energy, momentum and spin.
Electron capture: the nucleus takes in an inner (K shell) electron, \( p+e^-\to n+\nu \), and X-rays are given off as outer electrons fill the vacancy.
Gamma decay: an excited nucleus drops to a lower state and emits a photon; Z and A do not change.
Bookkeeping: an alpha emission changes (A, Z) by (−4, −2); beta-minus changes (A, Z) by (0, +1); beta-plus and electron capture change it by (0, −1).

Law of Radioactive Decay

The number of decays per second is proportional to the number of nuclei present: \( -\frac{dN}{dt}=\lambda N \), so \( N=N_0e^{-\lambda t} \). The decay constant \( \lambda \) is fixed for each nuclide.
Number decayed after time t: \( N_0(1-e^{-\lambda t}) \).
Half-life: \( T_{1/2}=\frac{\ln2}{\lambda}=\frac{0.693}{\lambda} \). After n half-lives (n = \( t/T_{1/2} \), not necessarily a whole number) the number left is \( N_0/2^n \).
Mean (average) life: \( \tau=\frac1\lambda=1.44\,T_{1/2} \).
Activity: \( A=\left|\frac{dN}{dt}\right|=\lambda N=A_0e^{-\lambda t} \). The SI unit is the becquerel (1 Bq = 1 decay per second); 1 curie = \( 3.7\times10^{10} \) decays per second. The activity left after n half-lives is \( A_0/2^n \).
If two nuclides have different half-lives and the same initial number, the one with the shorter half-life has the larger initial activity.

Production and Successive Decay

If a substance is produced at constant rate R and decays with constant \( \lambda \): \( \frac{dN}{dt}=R-\lambda N \), so \( N=\frac R\lambda(1-e^{-\lambda t}) \). The number grows toward the steady value \( R/\lambda \), at which production balances decay.
Successive decay \( A\xrightarrow{\lambda_1}B\xrightarrow{\lambda_2}C \): \( N_A=N_0e^{-\lambda_1t} \) and \( N_B=\frac{\lambda_1N_0}{\lambda_2-\lambda_1}\left(e^{-\lambda_1t}-e^{-\lambda_2t}\right) \). If \( \lambda_1\ll\lambda_2 \) the daughter reaches a state in which \( \lambda_1N_A\approx\lambda_2N_B \) (equilibrium of activities).

Nuclear Fission and the Reactor

In fission a heavy nucleus such as uranium-235, after absorbing a slow (thermal) neutron, splits into two medium fragments and releases two or three neutrons. About 200 MeV is released per fission, mostly as kinetic energy of the fragments; the mass lost is about 0.2 u. The released neutrons can cause further fissions, giving a chain reaction.
Reactor parts: fuel (enriched uranium); moderator (graphite, water or heavy water) to slow fast neutrons by collisions with light nuclei; control rods (cadmium or boron) that absorb neutrons; safety rods inserted in emergencies; and a coolant.
Multiplication factor \( K \) = fissions in one generation divided by fissions in the previous one. \( K=1 \): critical, steady power. \( K>1 \): supercritical, power rises exponentially. \( K<1 \): subcritical, reaction dies out.

Nuclear Fusion

In fusion, light nuclei join into a heavier one with release of energy, for example \( {}^2_1\mathrm H+{}^2_1\mathrm H\to{}^3_2\mathrm{He}+n+3.3\ \mathrm{MeV} \) and \( {}^2_1\mathrm H+{}^3_1\mathrm H\to{}^4_2\mathrm{He}+n+17.6\ \mathrm{MeV} \).
The nuclei must come within about 1 fm against their Coulomb repulsion, which requires temperatures of the order \( 10^{7} \) to \( 10^{9} \) K. Energy per unit mass of fuel is larger in fusion than in fission.
The proton-proton cycle in stars turns four hydrogen nuclei into one helium nucleus: \( 4\,{}^1_1\mathrm H\to{}^4_2\mathrm{He}+2e^++2\nu+26.7\ \mathrm{MeV} \) in total.

Worked Examples

Worked Example 1. Find the binding energy of an alpha particle. Masses: hydrogen atom 1.007826 u, neutron 1.008665 u, helium-4 atom 4.002603 u (1 u = 931 MeV).
Answer:
Mass defect \( =2(1.007826)+2(1.008665)-4.002603=0.030379 \) u.
Binding energy \( =0.030379\times931=28.3\ \text{MeV} \).

Worked Example 2. Find the Q-value of the beta-minus decay \( {}^{19}\mathrm O\to{}^{19}\mathrm F+e^-+\bar\nu \). Atomic masses: O-19 = 19.003576 u, F-19 = 18.998403 u.
Answer:
\( Q=(19.003576-18.998403)\times931=0.005173\times931=4.816\ \text{MeV} \).

Worked Example 3. A nucleus at rest emits an alpha particle in a decay with Q = 5.4 MeV and mass number A = 220 of the parent. Find the alpha energy.
Answer:
\( K_\alpha=\frac{A-4}{A}Q=\frac{216}{220}\times5.4=5.30\ \text{MeV} \).

Worked Example 4. What is the activity of 1 microgram of copper-64 (decay constant \( 1.516\times10^{-5}\ \mathrm{s^{-1}} \), atomic weight 63.5 g/mol)?
Answer:
Atoms \( =\frac{6\times10^{23}\times10^{-6}}{63.5}=9.45\times10^{15} \).
Activity \( \lambda N=1.516\times10^{-5}\times9.45\times10^{15}=1.43\times10^{11} \) per second \( =3.9 \) curie.

Worked Example 5. The half-life of a radionuclide is 20 hours. What fraction of the original activity is left after 40 hours?
Answer:
40 hours is 2 half-lives, so the fraction left is \( (1/2)^2=1/4 \).

Worked Example 6. A rock contains equal numbers of uranium-238 and lead-206 atoms, all the lead having come from the uranium (half-life \( 4.5\times10^9 \) y). Estimate the age of the rock.
Answer:
Equal numbers means half the uranium has decayed, so exactly one half-life has passed: \( 4.5\times10^9 \) years.

Worked Example 7. A radioisotope is made in a reactor at a constant rate R and decays with half-life 2.5 h. What is the steady number of nuclei if the final activity is \( 1.4\times10^{11} \) per second?
Answer:
At steady state activity equals R, so \( N=\frac{A}{\lambda} \) with \( \lambda=\frac{0.693}{2.5\times3600}=7.7\times10^{-5}\ \mathrm{s^{-1}} \).
\( N=\frac{1.4\times10^{11}}{7.7\times10^{-5}}=1.8\times10^{15} \).