Atomic Physics

⚛️Atomic Physics Unit 4 – Many–Electron Atoms

Many-electron atoms involve multiple electrons interacting with each other and the nucleus. This unit explores electron configurations, the Pauli exclusion principle, and Hund's rules, which govern how electrons occupy atomic orbitals. Atomic spectra, resulting from electron transitions, are examined along with coupling schemes that describe angular momentum combinations. These concepts are crucial for understanding spectroscopic applications in fields like astronomy and materials science.

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 (nn), angular momentum (ll), magnetic (mlm_l), and spin (msm_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: 1λ=RH(1n121n22)\frac{1}{\lambda} = R_H (\frac{1}{n_1^2} - \frac{1}{n_2^2}), where RHR_H is the Rydberg constant and n1n_1, n2n_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+1LJ^{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+1LJ^{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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.