7: Atomic Spectroscopy
- Page ID
- 6691
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 7.1: Introduction
- This page covers the discovery of spectral lines, focusing on their wavelengths and wavenumbers, with guidelines for measurement in nanometers and micrometers. It explains the relationships between wavelength, wavenumber, and frequency, and details atomic energy levels and ionization energies.
- 7.2: A Very Brief History of Spectroscopy
- This page outlines the evolution of spectroscopy beginning with Snel's law in 1621. It highlights significant developments such as Newton's light dispersion experiments, the discoveries of infrared and ultraviolet spectra, and Young’s wave theory. Key contributions from Fraunhofer, Kirchhoff, and Bunsen established foundational principles.
- 7.3: The Hydrogen Spectrum
- This page discusses J. J. Balmer's 1885 method for describing hydrogen's visible spectrum, establishing the Balmer series with a limit at 364.60 nm. It highlights the use of the Rydberg constant for calculating wavenumbers and mentions other identified series like Lyman and Paschen. Additionally, it notes that hydrogen-like atoms have similar spectra and addresses the fine structure of these lines, linked to the fine structure constant and its historical relevance to Sir Arthur Eddington.
- 7.4: The Bohr Model of Hydrogen-like Atoms
- This page examines Niels Bohr's 1913 atomic model, crucial for atomic theory and the hydrogen spectrum. It covers electron behavior in hydrogen-like atoms, including calculations for orbital radii, electron speeds, and energy levels, highlighting the Rydberg constant. Additionally, it discusses term values and wavelengths for hydrogen's energy levels, focusing on the Lyman and Balmer series.
- 7.5: One-dimensional Waves in a Stretched String
- This page explores wave behavior in a taut string to illustrate aspects of quantum mechanics. It describes how disturbances create waves governed by a second-order differential equation, resulting in standing waves due to fixed ends. These lead to discrete frequencies, tied to quantum numbers.
- 7.6: Vibrations of a Uniform Sphere
- This page covers the wave equation and wave function \(\Psi\) in three-dimensional spherical symmetry. It details the transformation to spherical coordinates, describes stationary solutions, and introduces quantum numbers \(n\), \(l\), and \(m\) influenced by boundary conditions. Furthermore, it illustrates the transition from classical to quantum mechanics and highlights the relevance of these concepts in various fields such as astrophysics and geophysics.
- 7.7: The Wave Nature of the Electron
- This page traces the development of wave-particle duality in physics, beginning with Barkla's 1906 x-ray polarization experiment and Bragg's 1913 x-ray diffraction. It covers Compton's 1919 demonstration of x-ray scattering's dual nature and de Broglie’s 1924 theory on electron duality, confirmed by Davisson and Germer's experiments.
- 7.8: Schrödinger's Equation
- This page covers the wave behavior of electrons and introduces key quantum mechanics equations, particularly Schrödinger's Equation in both time-dependent and time-independent forms. It derives these equations from periodic wave solutions and discusses kinetic and potential energy in quantum contexts. The page highlights how Schrödinger's Time-dependent Equation describes particle dynamics across different spatial dimensions, encapsulating essential principles of quantum behavior.
- 7.9: Solution of Schrödinger's Time-independent Equation for the Hydrogen Atom
- This page explores the Schrödinger equation for hydrogen-like atoms, focusing on spherical coordinates and the wavefunction's structure. It emphasizes the derived probability density and the significance of normalization for wavefunctions. Graphical representations clarify that electron positions are angular distributions rather than defined orbits.
- 7.10: Operators, Eigenfunctions and Eigenvalues
- This page covers fundamental concepts in quantum mechanics, focusing on Schrödinger’s Equation, commuting operators, and angular momentum. It highlights the Hamiltonian operator's role in defining eigenvalue problems related to wavefunctions and energy levels in systems like hydrogen atoms. The significance of commuting operators is explained, showing how they share common eigenfunctions.
- 7.11: Spin
- This page highlights the limitations of the Schrödinger equation in describing the hydrogen spectrum, emphasizing the need for four quantum numbers, including spin. It discusses the importance of the Stern-Gerlach experiment in illustrating intrinsic electron spin and magnetic moments. The page concludes by reinforcing the Pauli Exclusion Principle, which asserts that no two electrons can possess the same set of quantum numbers.
- 7.12: Electron Configurations
- This page covers the orbital angular momentum quantum numbers (\(l\)) for electrons, detailing the notations \(s\), \(p\), \(d\), and \(f\). It explains electron organization into principal shells (K, L, M, N) and connects this to the Pauli exclusion principle. An example using copper’s electron configuration illustrates how to derive quantum numbers for electrons in shells.
- 7.13: LS-coupling
- This page discusses angular momentum in atomic physics, highlighting the coupling of orbital (\(\textbf{L}\)) and spin (\(\textbf{S}\)) angular momentum to form total angular momentum (\(\textbf{J}\)). It explains the LS-coupling scheme's formulas for determining these magnitudes, noting its commonality in lighter elements and the need for alternative coupling schemes in heavier elements due to deviations.
- 7.14: States, Levels, Terms, Polyads, etc.
- This page covers electron configurations, particularly \(p^2\) states, detailing eigenfunctions and 15 possible quantum number combinations. It introduces terms and levels, defining key concepts like degeneracy and multiplicity while stressing clear notation.
- 7.15: Components, Lines, Multiplets, etc.
- This page discusses atomic transitions between energy levels, including processes like photoexcitation, absorption, and emission. It explains phenomena such as the Zeeman and Stark effects, which cause energy level splitting in magnetic and electric fields. Key terms in atomic spectroscopy are defined, along with notation for representing transitions. Additionally, it highlights the significance of selection rules that determine permissible transitions between defined atomic states.
- 7.16: Return to the Hydrogen Atom
- This page covers the electronic structure of hydrogen atoms, detailing energy levels and their statistical weights. It identifies the lowest configuration as \(1s\) with one term, \(^2 \text{S}_{1/2}\), followed by two terms in the \(L\)-shell and three in the \(M\)-shell. The page highlights the statistical weights of these shells and concludes by noting future discussions on selection rules related to hydrogen's spectral lines, specifically \(\text{H}\alpha\).
- 7.17: How to recognize LS-coupling
- This page covers \(LS\)-coupling in atomic physics, focusing on angular momentum interactions in light versus heavy atoms. It details allowed transitions between terms with the same \(S\) value and introduces Landé's Interval Rule related to the spin-orbit coupling coefficient \(a\). Additionally, it discusses term stability, separations, and provides examples of normal and inverted terms in elements such as magnesium and aluminum.
- 7.18: Hyperfine Structure
- This page explores hyperfine structure in atomic spectra, emphasizing the need for high-resolution detection techniques typically involving low temperatures and pressures. It details the impact of nuclear spin, influenced by nucleons' \(1/2\) spin, on angular momentum, denoted as \(\textbf{F}\). The page also underscores the importance of hyperfine structure in atomic phenomena, notably the 21-cm line of hydrogen, which corresponds to transitions between hyperfine levels.
- 7.19: Isotope effects
- This page examines the impact of isotopes on atomic structure, emphasizing how mass differences affect the center of mass and subsequently energy levels and spectral lines. Notably, significant isotope shifts are observed in heavy elements like lead and gold due to their substantial nuclear volumes.
- 7.20: Orbiting and Spinning Charges
- This page introduces the Zeeman effect and reviews classical mechanics and electromagnetism principles concerning orbiting and spinning electric charges. It calculates angular momentum and magnetic moment, elucidates magnetogyric ratio, and discusses potential energy of a magnetic moment in a magnetic field, detailing its angle dependence.
- 7.21: Zeeman effect
- This page covers the Zeeman effect, which is the splitting of atomic spectral lines in a magnetic field, a phenomenon first identified by P. Zeeman in 1896. It discusses the removal of degeneracy in energy levels due to angular momentum and highlights the Landé g-factor, which links angular momentum to magnetic moments.
- 7.22: Paschen-Back Effect
- This page explores atomic energy levels in magnetic fields, highlighting the differences between weak and strong fields. In weak fields, states are defined by total angular momentum \(J\), while in strong fields, angular momentum components \(L\) and \(S\) couple separately, resulting in the Paschen-Back effect. It examines interaction energies and angular momentum dependencies, featuring exercises that demonstrate the shift from Zeeman splitting to the Paschen-Back effect.
- 7.23: Zeeman effect with nuclear spin
- This page explores the effect of nuclear spin on atomic state splitting, detailing the interaction between electronic angular momentum \(J\) and nuclear spin \(I\) to yield total angular momentum \(F\). It covers hyperfine level spacing for a \(J=1\) and \(I=1\) system, the Zeeman effect in small magnetic fields, and the Paschen-Back effect at higher fields. The importance of high-resolution techniques for observing hyperfine structures is also highlighted.
- 7.24: Selection rules
- This page explores atomic multiplets and the selection rules that determine allowed and forbidden transitions, specifically focusing on electric dipole, magnetic dipole, and electric quadrupole transitions. It highlights how these rules are influenced by changes in quantum numbers and angular momentum, using neutral helium and calcium as examples.
- 7.25: Some forbidden lines worth knowing
- This page examines atomic states, contrasting stable ground states with less stable excited states that emit photons rapidly. It introduces metastable states where de-excitation is slower, enabling the observation of "forbidden" transitions in thin gases, relevant in astronomical contexts like auroras and emission nebulae.
- 7.26: Stark Effect
- This page discusses the Stark effect, which is the splitting of spectral lines in an electric field, noting its rarity in ionized stellar atmospheres. It highlights that even members of the Balmer series of hydrogen do not exhibit a central Stark component, leading to notable emission line anomalies. Furthermore, it addresses the impact of near-collisions between atoms that induce temporary electric dipoles resulting in pressure broadening of spectral lines.