Lecture 25: Quantum & nuclear physics

Overview of Quantum and Nuclear Physics

Focuses on electron shell structure, Pauli exclusion principle, atomic nucleus, nuclides, isotopes, and binding energy of atomic nucleus.

Nuclear Model of the Atom

Structure: Consists of a small, massive, positively charged nucleus containing protons and neutrons (nucleons) with negatively charged electrons orbiting around it in defined energy levels and orbital shells. This model is based on quantum mechanics, which describes the behavior of particles at the atomic and subatomic levels.

Electron Behavior: Unlike the classical planetary model, electrons do not follow fixed orbits but instead occupy orbitals—regions of space where there is a high probability of finding an electron at any given time. This is a consequence of quantum behavior, which influences the arrangement of electrons and their energy states.

Electron Shells

Energy Levels: The arrangement of electrons is defined by energy levels, with lower energy levels being closer to the nucleus. The first energy level can hold a maximum of 2 electrons, while subsequent levels can hold more, with the capacity increasing as distance from the nucleus increases.

Example: The hydrogen atom, with one electron, has its single electron in the first shell.

Increasing Protons and Electrons: As atomic number increases, the number of electrons in the higher energy levels increases according to the rules of electron configuration.

Example:

  • Helium: 2 protons, 2 electrons (both in 1st shell)

  • Lithium: 3 protons, 3 electrons (2 in 1st shell, 1 in 2nd shell)

  • Beryllium: 4 protons, 4 electrons (2 in 1st shell, 2 in 2nd shell)

Shell Capacity:
1st shell (K): Holds a maximum of 2 electrons
2nd shell (L): Holds a maximum of 8 electrons
3rd shell (M): Holds a maximum of 18 electrons
4th shell (N): Holds a maximum of 32 electrons

Subshells and Quantum Numbers

Principal Quantum Number (n): Indicates the primary energy level of an electron within an atom and determines its overall energy.

Sublevels: Each principal energy level beyond n=1 is divided into sublevels labeled s, p, d, and f, where each subshell can accommodate a specific number of electrons:

  • n=1 (1s): 1 subshell, can hold 2 electrons

  • n=2 (2s, 2p): 2s holds 2, 2p holds 6 (Total: 8)

  • n=3 (3s, 3p, 3d): 3s holds 2, 3p holds 6, 3d holds 10 (Total: 18)

  • n=4 (4s, 4p, 4d, 4f): 4s holds 2, 4p holds 6, 4d holds 10, 4f holds 14 (Total: 32)

Pauli Exclusion Principle

Definition: The Pauli exclusion principle states that no two fermions (e.g., electrons) can occupy the same quantum state within a quantum system simultaneously. This principle is fundamental to the structure of atoms.

Impact on Electron Configuration: It limits the maximum number of electrons that can occupy a subshell and dictates how electrons fill the available energy levels and subshells. Electrons tend to fill from the lowest energy level upwards, minimizing energy within the atom.

The Atomic Nucleus

Composition: The atomic nucleus is made up of protons and neutrons, collectively known as nucleons. Protons are positively charged, while neutrons carry no charge. The number of protons determines the atomic number and identity of the element.

Forces Involved:

  • Strong Nuclear Force: This is a fundamental force that attracts nucleons toward each other, effectively overcoming the electrostatic repulsion caused by like-charged protons within the nucleus.

  • Neutron Role: Neutrons play a crucial role in stabilizing the atomic nucleus. Their presence dilutes the repulsion among protons and contributes to the overall nuclear stability.

Nuclides and Isotopes

Nuclide Definition: A nuclide is a particular isotope of an element characterized by a specific number of protons and neutrons. Each nuclide has its own unique energy state.

Types:

  • Stable nuclides (e.g., carbon-12), which do not undergo radioactive decay.

  • Radioactive nuclides (e.g., uranium-238) that undergo decay at different rates, represented by their half-lives.

Isotopes: Isotopes are variants of a given element that have the same number of protons but different numbers of neutrons. For example, carbon has isotopes such as carbon-12, carbon-13, and carbon-14, each differing by the number of neutrons present in the nucleus.

Nuclear Notation

Symbols Used: Nuclear notation summarizes essential information about nuclides using the following symbols:

  • Elemental symbol (e.g., Ba for barium).

  • Atomic number (representing the number of protons).

  • Mass number (total count of protons plus neutrons).

Notation for particles:

  • Proton: small p, Z=1, A=1

  • Neutron: small n, Z=0, A=1

  • Electron: small e, Z=-1, A=0

Binding Energy

Definition: Binding energy is defined as the energy that holds the nucleus together and corresponds to the mass decrement that occurs when nucleons bind together to form a nucleus. It reflects the work necessary to separate individual nucleons in a nucleus.

Calculation: The binding energy can be calculated by taking the total mass of the constituent nucleons before they are bound together and subtracting the actual mass of the nucleus formed.

Example: For carbon-12, the derived binding energy is approximately 92.2 MeV, which yields a binding energy per nucleon of about 7.68 MeV. This metric signifies the stability and energy characteristics of the nuclide.

Stability and Energy Release

Graph of Binding Energy: A significant characteristic of nuclear physics is that light nuclei can fuse to create heavier nuclei, while heavy nuclei tend to undergo fission, splitting into lighter components. This process releases vast amounts of energy due to the conversion of mass into energy as defined by Einstein's equation E=mc².

Iron (atomic number 56) stands out as having the highest stability peak in the binding energy curve, making it a significant element in stellar nucleosynthesis.

Practice Questions

  1. Electron Configuration of Nitrogen: 1s² 2s² 2p³

  2. Reason for Maximum 2 Electrons in 1st Shell: Due to the Pauli exclusion principle, which restricts identical fermions from occupying the same quantum state.

  3. General Pattern of Nuclear Structure: Ratio of protons to neutrons decreases with increasing mass as the stability of the nucleus is affected by the number of neutrons.

  4. Energy from Fission of Uranium-235: The mass difference between the initial uranium nucleus and the resulting products leads to the release of energy according to mass-energy equivalence.

  5. Binding Energy of Deuterium: The binding energy is approximately 1.11 MeV per nucleon, indicating the nuclear stability achieved through the combination of one proton and one neutron in deuterium, a hydrogen isotope.