7.1 Discrete Energy and Radioactivity

Discrete energy:

Emission spectrum represents different possible wavelengths of light emitted by an atom.

• When gas at low pressure is subjected to a strong electric field, it emits light at discrete wavelengths.

• The emission spectrum (e.g., hydrogen, helium, mercury) comprises lines at specific wavelengths, representing photon emissions during electron transitions between energy levels.

• Emission spectrum: Series of bright lines representing the wavelengths that can be emitted by an atom.

• From the excited state, the electron will immediately (within nanoseconds) make a transition down to one of the available lower energy states.This process is called relaxation.

• , h is planck's constant, is the wavelength emitted during relaxation and c is the speed of light, whereas E is the energy released.

• This means that the light that is transmitted through the gas will be missing the photons that have been absorbed.This gives rise to absorption spectra

• Radioactivity: Spontaneous emission of particles and energy from an unstable nucleus.

• Discovered by Henri Becquerel, Marie Skłodowska-Curie (1867–1934), and Pierre Curie (1859–1906).

  • Alpha particles: Helium nucleus emitted during alpha decay.

  • Beta particles: Electrons or positrons emitted during beta decay.

  • Gamma rays: High-frequency electromagnetic radiation from nucleus transitions.

Nuclear transmutation: Transformation of one element to another through nuclear reactions, such as alpha particle collision with nitrogen to produce oxygen and a proton.

Nuclear fission:

Splitting of a heavy nucleus into lighter nuclei, with the release of energy.

Example: Absorption of a neutron by uranium-235, resulting in uranium-236, which then fissions into krypton, barium, and more neutrons.

Chain reaction: Self-sustaining fission process due to released neutrons inducing further reactions.

Critical mass: Minimum mass of fissile material needed to maintain a chain reaction.

Nuclear fusion:

Joining of light nuclei to form a heavier nucleus, releasing energy.

Occurs in stars and hydrogen bombs, but controlled fusion for energy production is still a challenge.

• Nuclear structure: Constituents of a nucleus (protons and neutrons) and their organization.

  • Atomic (proton) number (Z): Number of protons in a nucleus.

  • Mass (nucleon) number (A): Sum of protons and neutrons in the nucleus.

  • Neutrons (N): Calculated as N = A - Z.

Isotopes: Atoms with the same number of protons but different numbers of neutrons. Physical properties vary, but chemical properties are identical due to the same number of electrons.

Radioactive isotopes: Exhibit spontaneous radioactive decay.

Radioactive decay:

Unstable nuclei emit particles and energy spontaneously.

Types of decay: alpha (α), beta (β), and gamma (γ) radiation.

Alpha decay: Emission of an alpha particle (helium nucleus).

Beta decay: Neutron transforms into a proton emitting an electron (beta-minus decay) or proton transforms into a neutron emitting a positron (beta-plus decay).

Gamma decay: Emission of a gamma ray, no change in the nucleus's atomic number or mass number.

Half-Life & Probability:

Half-life (t_1/2): The time required for half the quantity of a radioactive substance to undergo decay. A measure of the stability of a radioactive isotope; shorter half-life indicates a more unstable isotope. Determines the rate at which a sample loses its radioactivity.

Exponential decay: The number of undecayed nuclei decreases exponentially over time.

Probability in decay:

Each nucleus has a constant probability of decaying in a given time interval, independent of time.

After one half-life, the probability that a nucleus has not decayed is 50%.

Multiple half-lives follow a predictable pattern: after n half-lives, the fraction remaining is

Decay Series

Radioactive decay series: A sequence of decay events from a parent radionuclide to stable daughter isotopes.

Example: The decay series of uranium-238 to lead-206 involves multiple alpha and beta decays.

Each step in the series has its own characteristic half-life and decay mode.

The Law of Radioactive Decay

Radioactive decay law: States that the activity (rate of decay) of a radioactive sample is proportional to the number of undecayed nuclei present at any time.

Mathematically expressed as

, where N is the amount of undecayed nuclei.

7.2 Nuclear Reactions

Transmutation and energy release:

Unified atomic mass unit (u): A standard unit of mass that quantifies mass on an atomic or molecular scale.

1 u is defined as one twelfth the mass of a carbon-12 atom, approximately

1.660539×10^-27 kilograms.

Mass Defect and Binding Energy

Mass defect (Δ): The difference between the mass of the completely separated nucleons and the mass of the nucleus.

Occurs because mass is converted into binding energy when the nucleus forms.

Formula:

Where is Z is the number of protons, N is the number of neutrons and and is the mass of the proton and mass of the neutron respectively, whereas is the actual mass of the nucleus.

Binding energy: The energy required to disassemble a nucleus into its individual protons and neutrons.

Calculated using Einstein's equation

m is mass defect often given as a lower case delta, and c is the speed of light.

The Binding Energy Curve:

The binding energy per nucleon varies with the nucleon number and has a peak at iron-56, indicating the greatest stability.

Light nuclei (up to iron) gain stability through fusion, while heavy nuclei (beyond iron) gain stability through fission.

Energy Released in Decay

Nuclear fission: A heavy nucleus splits into two smaller nuclei, releasing a large amount of energy.

Example: Uranium-235 undergo fission after capturing a neutron.

Energy released is due to the conversion of mass defect into energy, typically measured in mega-electron volts (MeV).

Nuclear fusion: Lighter nuclei combine to form a heavier nucleus, releasing energy.

Example: Fusion of deuterium and tritium to form helium-4.

Requires high temperatures and pressures to overcome electrostatic repulsion, with the sun being a natural fusion reactor.

7.3 The Structure of Matter

Particle physics:

Investigates fundamental building blocks of matter (quarks and leptons) and their interactions.

Rutherford experiment: Revealed the nucleus and led to the planetary model of the atom.

Fundamental particles:

Quarks: Six types ('flavors') with different properties, combining to form particles like protons and neutrons.

Leptons: Include electrons, neutrinos, and their anti-particles, not subject to the strong interaction.

Exchange particles: Mediate fundamental forces (e.g., photons for electromagnetic force).

Nuclear Forces and Particles:

Strong nuclear force: Binds quarks within protons and neutrons, and these nucleons within the nucleus.

Alpha, beta, and gamma decay: Processes by which unstable nuclei release particles and energy.

The Higgs boson: Particle associated with the Higgs field, which gives mass to other particles in the Standard Model.

Exchange Particles and Fundamental Forces:

Electromagnetic interactions: Mediated by photons.

Weak interactions: Involve W and Z bosons, responsible for processes like beta decay.

Strong interactions: Governed by gluons, binding quarks together within nucleons.

Gravitational interactions: Attributed to gravitons, though not yet experimentally confirmed.

Conservation Laws in Particle Physics

Baryon number: Conserved in nuclear reactions; associated with quarks and baryons.

Lepton number: Conserved for electrons, muons, and their respective neutrinos.

Strangeness: Quantum number conserved in strong interactions, may change in weak interactions.

Electric charge: Conserved in all types of interactions.

Feynman Diagrams:-

Visual representations: Depict particle interactions, with particles as lines and interactions as vertices.

Interaction vertices: Show the exchange of force carriers like photons and W/Z bosons.

Important for calculations: Simplify understanding of complex interactions in quantum field theory.

Exam Tips

• Be prepared to apply knowledge of discrete energy levels and transitions to solve problems.

• Remember to convert eV to joules for energy-related calculations.

• Understand the significance of the binding energy curve and its implications for nuclear stability.

• Be familiar with Feynman diagrams to represent particle interactions and decays.