⚛️Atomic Physics Unit 4 – Many–Electron Atoms

Key Concepts

• Many-electron atoms contain multiple electrons interacting with each other and the nucleus
• Electron configuration describes the arrangement of electrons in an atom's orbitals
• Pauli exclusion principle states that no two electrons in an atom can have the same quantum numbers
  • Limits the number of electrons that can occupy each orbital
• Hund's rules determine the ground state electron configuration of an atom
  • Electrons maximize total spin, then total orbital angular momentum, and finally occupy orbitals of lowest energy
• Atomic spectra result from transitions between different electronic states in an atom
  • Emission spectra occur when electrons transition from higher to lower energy states (hydrogen spectrum)
  • Absorption spectra happen when electrons absorb energy and move to higher energy states
• Coupling schemes describe how angular momenta of individual electrons combine to form the total angular momentum of the atom
  • LS coupling (Russell-Saunders coupling) and jj coupling are two common schemes
• Applications of many-electron atom concepts in spectroscopy enable the identification and analysis of elements and compounds
  • Used in fields such as astronomy, materials science, and analytical chemistry

Electron Configuration

• Electron configuration represents the distribution of electrons in an atom's orbitals
• Orbitals are characterized by quantum numbers: principal ($n$), angular momentum ($l$), magnetic ($m_l$), and spin ($m_s$)
• Electrons fill orbitals in order of increasing energy, following the Aufbau principle
  • 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p
• Shorthand notation represents electron configuration using noble gas cores and valence electrons
  • Neon (1s²2s²2p⁶) can be abbreviated as [Ne]
  • Sodium (1s²2s²2p⁶3s¹) is written as [Ne] 3s¹
• Exceptions to the Aufbau principle occur due to electron-electron interactions and relativistic effects
  • Copper: [Ar] 3d¹⁰4s¹ instead of [Ar] 3d⁹4s²
  • Chromium: [Ar] 3d⁵4s¹ instead of [Ar] 3d⁴4s²

Pauli Exclusion Principle

• The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers
• Consequence of the antisymmetric nature of the electron wavefunction under particle exchange
• Limits the number of electrons that can occupy each orbital
  • s orbitals (l=0) can hold up to 2 electrons with opposite spins
  • p orbitals (l=1) can hold up to 6 electrons, 2 in each of the three ml states (-1, 0, +1)
• Responsible for the stability of matter and the periodic table's structure
• Explains the shell structure of atoms and the existence of chemical elements with distinct properties
• Violation of the Pauli exclusion principle would lead to the collapse of atoms and the breakdown of chemistry as we know it

Hund's Rules

• Hund's rules determine the ground state electron configuration of an atom when there are multiple degenerate orbitals
• First rule: Maximize total spin angular momentum (S)
  • Electrons occupy orbitals singly with parallel spins before pairing up
• Second rule: Maximize total orbital angular momentum (L)
  • Electrons distribute themselves to maximize the total orbital angular momentum
• Third rule: Minimize total angular momentum (J) for less than half-filled shells, and maximize J for more than half-filled shells
  • J = |L - S| for less than half-filled shells
  • J = L + S for more than half-filled shells
• Example: Carbon (1s²2s²2p²) has two unpaired electrons in separate 2p orbitals with parallel spins
• Hund's rules result from the interplay between electron-electron repulsion and spin-orbit coupling
• Help predict the ground state term symbol and magnetic properties of atoms

Atomic Spectra

• Atomic spectra result from transitions between different electronic states in an atom
• Emission spectra occur when electrons transition from higher to lower energy states
  • Hydrogen emission spectrum consists of distinct lines in the visible, UV, and IR regions (Lyman, Balmer, Paschen series)
• Absorption spectra happen when electrons absorb energy and move to higher energy states
  • Dark lines in the solar spectrum correspond to absorption by elements in the Sun's atmosphere (Fraunhofer lines)
• Energy differences between electronic states determine the wavelengths of spectral lines
  • Rydberg formula: $\frac{1}{\lambda} = R_H (\frac{1}{n_1^2} - \frac{1}{n_2^2})$, where $R_H$ is the Rydberg constant and $n_1$, $n_2$ are principal quantum numbers
• Selection rules govern allowed transitions based on changes in quantum numbers
  • Electric dipole transitions: Δl = ±1, Δml = 0, ±1, ΔS = 0
• Fine structure results from spin-orbit coupling and relativistic corrections
  • Splitting of spectral lines into closely spaced components (sodium D lines)
• Hyperfine structure arises from the interaction between the electron and nuclear magnetic moments
  • Further splitting of fine structure components (21 cm hydrogen line)

Coupling Schemes

• Coupling schemes describe how angular momenta of individual electrons combine to form the total angular momentum of the atom
• LS coupling (Russell-Saunders coupling) applies when spin-orbit interaction is weak compared to electron-electron interaction
  • Orbital angular momenta of electrons couple to form total orbital angular momentum L
  • Spin angular momenta of electrons couple to form total spin angular momentum S
  • L and S then couple to form total angular momentum J
  • Term symbols: $^{2S+1}L_J$, where L is represented by letters (S, P, D, F, ...)
• jj coupling applies when spin-orbit interaction is strong compared to electron-electron interaction
  • Orbital and spin angular momenta of each electron couple to form individual total angular momenta j
  • Individual j's then couple to form total angular momentum J
  • Relevant for heavy atoms and excited states
• Intermediate coupling occurs when spin-orbit and electron-electron interactions are comparable
  • Requires a more complex mathematical treatment
• Coupling schemes help interpret atomic spectra and predict the energy levels and transitions in atoms

Applications in Spectroscopy

• Many-electron atom concepts find extensive applications in various spectroscopic techniques
• Atomic absorption spectroscopy (AAS) uses the absorption of light by atoms to determine the concentration of elements in a sample
  • Widely used in analytical chemistry for quantitative analysis of metals
• Atomic emission spectroscopy (AES) analyzes the light emitted by atoms to identify and quantify elements
  • Employed in industries such as metallurgy, environmental monitoring, and forensic science
• Laser-induced breakdown spectroscopy (LIBS) uses a high-power laser to create a plasma and measure the atomic emission spectra
  • Enables rapid, in-situ analysis of solid, liquid, and gaseous samples
  • Applications in space exploration, cultural heritage studies, and biomedical research
• Stellar spectroscopy relies on atomic spectra to determine the composition, temperature, and velocity of stars and galaxies
  • Crucial for understanding the evolution and structure of the universe
• X-ray fluorescence (XRF) spectroscopy probes the emission of characteristic X-rays from atoms excited by high-energy radiation
  • Non-destructive technique for elemental analysis of materials, artworks, and archaeological artifacts

Problem-Solving Techniques

• Solving problems related to many-electron atoms requires a systematic approach and application of key concepts
• Determine the electron configuration of the atom using the Aufbau principle, Pauli exclusion principle, and Hund's rules
  • Identify the number of electrons and the order of orbital filling
  • Distribute electrons in orbitals according to Hund's rules
• Use term symbols to represent the electronic states of the atom
  • Determine S, L, and J based on the electron configuration and coupling scheme
  • Follow the conventions for labeling terms (e.g., $^{2S+1}L_J$)
• Apply selection rules to identify allowed transitions between electronic states
  • Consider the changes in quantum numbers (Δl, Δml, ΔS) for electric dipole transitions
  • Use parity considerations for centrosymmetric atoms (Laporte rule)
• Interpret atomic spectra by relating spectral lines to transitions between energy levels
  • Use the Rydberg formula to calculate the wavelengths of transitions
  • Identify the series (Lyman, Balmer, Paschen) based on the principal quantum numbers involved
• Utilize the concepts of fine and hyperfine structure to explain the splitting of spectral lines
  • Consider the effects of spin-orbit coupling and electron-nuclear interactions
  • Calculate the energy shifts and splittings using appropriate formulas (e.g., fine structure constant, hyperfine coupling constants)
• Analyze the implications of many-electron atom concepts in practical applications
  • Relate the principles to spectroscopic techniques used in various fields
  • Discuss how atomic spectra provide insights into the properties and behavior of atoms and molecules