The Atomic Nucleus: Masses and Stability - Detailed Notes

1. Nuclear Forces:

• Strong Nuclear Force:

• Exchange of mesons like pions(pi)

• Saturation property: strong nuclear force is strong only at very short distances and becomes negligible beyond a certain distance.

• Charge independence: the strong nuclear force is independent of the charges of the interacting nucleons.

• Isospin: concept used to describe the charge states of nucleons.

• Weak Nuclear Force:

• Charged and neutral currents: two types of weak interactions, mediated by W and Z bosons.

• W bosons: mediate the charged weak interactions responsible for beta decay.

• Z bosons: mediate the neutral weak interactions responsible for neutrino scattering and electron-positron annihilation.

2. Nuclear Models:

• Liquid Drop Model:

• Semi-empirical mass formula: provides an empirical relationship between the mass of a nucleus and its number of protons and neutrons.

• Binding energy: energy required to separate all the nucleons in a nucleus from each other.

• Explanation for odd-even rule: odd-numbered nuclei are less stable than even-numbered nuclei due to the pairing of nucleons in the nucleus.

• Limiting value of Z for stability: the maximum number of protons that a nucleus can have for a given number of neutrons before becoming unstable.

• Shell Model:

• Quantum mechanical model of the nucleus, based on the idea of nucleons occupying specific energy levels or shells within the nucleus.

• Nucleon: a general term for a proton or a neutron.

• Magic numbers: special numbers of nucleons (2, 8, 20, 28, 50, 82, 126) that correspond to particularly stable configurations.

• Spin-orbit coupling: interaction between the spin of a nucleon and its orbital motion around the nucleus, which affects the energy levels of nucleons.

• Collective Model:

• Describes the nucleus as a rotating or vibrating object.

• Rotational and vibrational modes of excitation: different ways in which the nucleus can rotate or vibrate, resulting in excited states.

• Nuclear deformation: change in the shape of the nucleus from a perfect sphere, which can occur due to rotation or vibration.

• Moments of inertia: measure of the resistance of a rotating nucleus to changes in its angular velocity.

• Fermi Gas Model:

• Treats the nucleons in a nucleus as non-interacting particles that obey Fermi statistics.

• Concept of Fermi energy: the highest energy level that a nucleon can occupy at absolute zero temperature.

• Pauli’s exclusion principle: no two nucleons can occupy the same quantum state at the same time.

3. Nuclear Stability:

• Mass Defect and Binding Energy:

• Mass defect: the difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons.

• Binding energy: the energy equivalent of the mass defect, representing the energy required to separate all the nucleons in a nucleus from each other.

• Beta Decay:

• Types of beta decay (β+, β-): emission of a positron (β+) or an electron (β-) from the nucleus to convert a proton into a neutron or vice versa.

• Energy released: the energy released in beta decay is equal to the difference between the mass of the parent nucleus and the mass of the daughter nucleus, minus the mass of the emitted particle.

• Q-value: the maximum kinetic energy of the emitted particle in beta decay.

• Alpha Decay:

• Emission of an alpha particle (consisting of two protons and two neutrons) from the nucleus.

• Q-value: the energy released in alpha decay is equal to the difference between the mass of the parent nucleus and the mass of the daughter nucleus, minus the mass of the alpha particle.

• Gamow’s theory: explains alpha decay as the tunneling of the alpha particle through the potential barrier created by the strong nuclear force.

• Gamma Decay:

• Decay scheme: representation of the sequence of transitions between energy levels in a nucleus, involving the emission of gamma rays.

• Selection rules: restrictions on the transitions between energy levels that can occur, based on the conservation of angular momentum and parity.

• Internal conversion: process in which the energy of an excited nucleus is transferred to one of its electrons, causing the electron to be emitted.

• Emission of positrons: occurs in certain cases, when an electron-positron pair is created from the energy of the excited nucleus.

4. Nuclear Reactions:

• Types of Nuclear Reactions:

• Nuclear fission: process in which a heavy nucleus splits into two or more smaller nuclei, releasing energy.

• Nuclear fusion: process in which two or more light nuclei combine to form a heavier nucleus, releasing energy.

• Nuclear transmutations: processes in which the composition of a nucleus is changed, such as by the absorption of a neutron or the emission of an alpha particle.

• Conservation Laws:

• Energy: the total energy before and after a nuclear reaction must be the same.

• Mass number: the total number of protons and neutrons must be the same before and after a nuclear reaction.

• Charge: the total electric charge must be the same before and after a nuclear reaction.

• Momentum: the total momentum must be the same before and after a nuclear reaction.

• Spin: the total spin must be the same before and after a nuclear reaction.

• Cross-Section:

• Definition: a measure of the probability of a nuclear reaction occurring when a beam of particles interacts with a target.

• Types: various types of cross-sections, such as total cross-section, elastic cross-section, inelastic cross-section, absorption cross-section, scattering cross-section.

• Units: typically expressed in barns (1 barn = 10^-28 m^2) or millibarns (1 millibarn = 10^-31 m^2).

5. Applications of Nuclear Physics:

• Nuclear Power Plants:

• Fission reactors: use the fission process to generate heat, which is then used to produce steam and generate electricity.

• Types of fission reactors: pressurized water reactors (PWRs), boiling water reactors (BWRs), and gas-cooled reactors (GCRs).

• Advantages: provide a reliable and efficient source of energy, with low operating costs.

• Disadvantages: concerns related to safety, radioactive waste disposal, and proliferation.

• Breeder reactor: a type of reactor that produces more fissile material (such as plutonium-239) than it consumes.

• Nuclear Medicine:

• Radioisotopes in medical diagnosis: use of radioactive isotopes to diagnose various medical conditions, such as thyroid disorders and bone diseases.

• PET (Positron Emission Tomography): imaging technique that uses positron-emitting radioisotopes to produce three-dimensional images of the body’s metabolic activity.

• MRI (Magnetic Resonance Imaging): imaging technique that uses magnetic fields and radio waves to produce detailed images of the body’s internal structures.

• Radiotherapy: use of ionizing radiation to treat cancer and other diseases.

• Radiopharmaceuticals: radioactive substances used in medical diagnosis and therapy.

• Nuclear Technology:

• Radiocarbon dating: technique used to determine the age of organic materials by measuring the amount of radioactive carbon-14 present.

• Neutron radiography: technique that uses neutrons to produce images of objects, allowing the detection of hidden defects or structures.

• Food irradiation: process of treating food with ionizing radiation to extend its shelf life and eliminate harmful microorganisms.

• Smoke detectors: use a radioactive isotope to detect smoke particles and trigger an alarm in case of fire.

References:

• NCERT Physics, Class 11: Chapters 11-13 and 15
• NCERT Physics, Class 12: Chapters 13 and 14
• Concepts of Physics, Volume 2, by H. C. Verma
• Physics for Scientists and Engineers, Volume 2, by R. A. Serway and J. W. Jewett
• Nuclear Physics, by S. N. Goshal