🪐Principles of Physics IV Unit 5 – Atomic Structure and Spectra

Fundamental Concepts of Atomic Structure

• Atoms are the basic building blocks of matter consisting of a dense nucleus surrounded by electrons
• The nucleus contains positively charged protons and neutral neutrons held together by the strong nuclear force
• Electrons are negatively charged particles that orbit the nucleus in discrete energy levels or shells
• Atoms are electrically neutral with an equal number of protons and electrons
• The atomic number represents the number of protons in an atom's nucleus and determines its chemical properties
• Isotopes are atoms of the same element with different numbers of neutrons in their nuclei
• The mass number is the sum of the number of protons and neutrons in an atom's nucleus
• Atoms can form chemical bonds by sharing, gaining, or losing electrons to achieve a stable electronic configuration

Historical Models of the Atom

• Dalton's atomic theory proposed that atoms were indivisible and indestructible particles (early 19th century)
• Thomson's plum pudding model suggested that atoms were positively charged spheres with embedded negative electrons (late 19th century)
• Rutherford's gold foil experiment demonstrated that atoms have a small, dense, positively charged nucleus surrounded by electrons (early 20th century)
  • Alpha particles were fired at a thin gold foil
  • Most particles passed through undeflected, but some were deflected at large angles or even backscattered
• Bohr's atomic model introduced the concept of stationary electron orbits and energy levels (1913)
  • Electrons can only occupy specific orbits with fixed energies
  • Electrons can transition between orbits by absorbing or emitting photons with specific energies
• The Bohr model successfully explained the hydrogen atom's spectrum but failed for more complex atoms

Quantum Mechanical Model

• The quantum mechanical model describes atoms using wave functions and probability distributions
• Electrons are treated as wave-particle duality entities with properties like position and momentum described by probability distributions
• The Schrödinger equation is used to calculate the wave functions and energy levels of electrons in an atom
  • $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat H \Psi(\mathbf{r},t)$
  • $\Psi(\mathbf{r},t)$ is the wave function, $\hat H$ is the Hamiltonian operator, and $\hbar$ is the reduced Planck's constant
• The square of the wave function $|\Psi(\mathbf{r},t)|^2$ represents the probability density of finding an electron at a given position and time
• The Heisenberg uncertainty principle states that the position and momentum of an electron cannot be simultaneously determined with arbitrary precision
  • $\Delta x \Delta p \geq \frac{\hbar}{2}$
• The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers

Electron Configuration and Orbitals

• Electron configuration describes the distribution of electrons in an atom's orbitals
• Orbitals are regions in space where electrons are most likely to be found
• Orbitals are characterized by four quantum numbers: principal ($n$), angular momentum ($l$), magnetic ($m_l$), and spin ($m_s$)
  • The principal quantum number $n$ determines the energy level and size of the orbital
  • The angular momentum quantum number $l$ determines the shape of the orbital (s, p, d, f)
  • The magnetic quantum number $m_l$ determines the orientation of the orbital in space
  • The spin quantum number $m_s$ describes the intrinsic angular momentum of the electron (up or down)
• Electrons fill orbitals in order of increasing energy following Hund's rule and the Aufbau principle
• Hund's rule states that electrons occupy orbitals of the same energy singly before pairing up
• The Aufbau principle states that electrons fill orbitals in order of increasing energy (1s, 2s, 2p, 3s, 3p, 4s, 3d, ...)

Energy Levels and Transitions

• Electrons in atoms occupy discrete energy levels corresponding to specific orbitals
• The energy of an electron depends on its principal and angular momentum quantum numbers
• Electrons can transition between energy levels by absorbing or emitting photons with specific energies
  • The energy of a photon is given by $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency of the photon
• The Rydberg formula relates the wavelength of a photon to the energy difference between two levels in hydrogen-like atoms
  • $\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$
  • $\lambda$ is the wavelength, $R$ is the Rydberg constant, and $n_1$ and $n_2$ are the principal quantum numbers of the initial and final states
• Selection rules govern the allowed transitions between energy levels based on changes in quantum numbers
  • The principal quantum number must change ($\Delta n \neq 0$)
  • The angular momentum quantum number must change by $\pm 1$ ($\Delta l = \pm 1$)
  • The spin quantum number must not change ($\Delta m_s = 0$)

Atomic Spectra: Emission and Absorption

• Atomic spectra are the characteristic wavelengths of light emitted or absorbed by atoms
• Emission spectra are produced when electrons transition from higher to lower energy levels, emitting photons
  • Emission spectra appear as bright lines against a dark background
  • Examples include the hydrogen Balmer series (visible) and Lyman series (ultraviolet)
• Absorption spectra are produced when electrons absorb photons and transition from lower to higher energy levels
  • Absorption spectra appear as dark lines against a bright background
  • Examples include the Fraunhofer lines in the solar spectrum
• The wavelengths of spectral lines are unique to each element and can be used for identification
• Fine structure and hyperfine structure in spectra arise from interactions between electrons and the nucleus
  • Fine structure is caused by spin-orbit coupling and relativistic effects
  • Hyperfine structure is caused by the interaction between the electron and nuclear magnetic moments

Spectroscopic Techniques and Applications

• Spectroscopy is the study of the interaction between matter and electromagnetic radiation
• Atomic absorption spectroscopy (AAS) measures the absorption of light by atoms to determine elemental composition and concentration
  • A sample is vaporized and exposed to light from a hollow cathode lamp
  • The amount of light absorbed is proportional to the concentration of the element in the sample
• Atomic emission spectroscopy (AES) measures the light emitted by atoms to determine elemental composition and concentration
  • A sample is excited by a flame, plasma, or electrical discharge
  • The wavelengths and intensities of the emitted light are analyzed to identify and quantify the elements present
• Laser-induced breakdown spectroscopy (LIBS) uses a high-energy laser pulse to create a plasma and analyze the emitted light
  • LIBS allows for rapid, in-situ analysis of solid, liquid, and gaseous samples
  • Applications include environmental monitoring, forensic analysis, and space exploration
• X-ray fluorescence (XRF) spectroscopy uses high-energy X-rays to excite inner shell electrons and measure the emitted fluorescent X-rays
  • XRF is used for non-destructive elemental analysis of materials, artworks, and archaeological artifacts

Advanced Topics and Current Research

• Quantum computing utilizes the principles of quantum mechanics to develop powerful computational systems
  • Quantum bits (qubits) can exist in superposition states and exhibit quantum entanglement
  • Quantum algorithms (Shor's, Grover's) can solve certain problems much faster than classical computers
• Quantum cryptography uses the principles of quantum mechanics to ensure secure communication
  • Quantum key distribution (QKD) protocols (BB84, E91) allow for the secure exchange of cryptographic keys
  • Any attempt to intercept or measure the quantum states alters them, revealing the presence of an eavesdropper
• Attosecond science studies ultrafast electron dynamics on the attosecond ($10^{-18}$ s) timescale
  • Attosecond laser pulses can probe and control electron motion in atoms and molecules
  • Applications include studying chemical reactions, electron transfer, and photosynthesis
• Quantum sensing exploits the sensitivity of quantum systems to external perturbations for high-precision measurements
  • Atomic clocks use the transitions between hyperfine levels in atoms (cesium, ytterbium) for ultra-precise timekeeping
  • Quantum magnetometers use the spin states of atoms (rubidium, potassium) to detect weak magnetic fields
  • Quantum sensors have applications in navigation, medical imaging, and fundamental physics research