4: Spectroscopy
- Page ID
- 540242
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Spectroscopic Selection Rules
- Quantum systems can transition between energy states by absorbing or emitting electromagnetic radiation. This section discusses how we calculate the probability of such a transition and how through the use of symmetry of the eigenfunctions and the transition operator we can determine selection rules. The particle in a box and hydrogen-like atoms are used as an example of how this can be used to predict which transitions are observed.
- 4.2: Rotational Spectroscopy of Diatomic Molecules
- The permanent electric dipole moments of polar molecules couple to the electric field of electromagnetic radiation to induce transitions between the rotational states of the molecules. The energies that are associated with these transitions are detected in the far infrared and microwave regions of the spectrum. The selection rules for the rotational transitions are derived from the transition moment integral by using the spherical harmonic functions and the appropriate dipole moment operator.
- 4.3: Rotational Spectra of Polyatomic Molecules
- This page provides an overview of the angular motion and rotational dynamics of diatomic and polyatomic molecules. It begins with the Schrödinger equation for rigid diatomic molecules and introduces rotational energy levels governed by quantum number \(J\). The text further explores polyatomic molecules using an inertia tensor and discusses spherical tops with equal principal moments of inertia.
- 4.4: Harmonic Oscillator
- The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, AC circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter.
- 4.5: Hermite Polynomials are either Even or Odd Functions
- This page explores Hermite polynomials, focusing on their orthogonality, symmetry, and applications in quantum mechanics, especially as solutions for harmonic oscillators. It explains their historical context, definitions, and recurrence relations, alongside the significance of even and odd functions. The text illustrates the orthogonality of specific Hermite polynomials using integral methods over symmetric intervals, emphasizing how the product of even and odd functions affects their integrals.
- 4.6: The Harmonic Oscillator and Infrared Spectra
- This page explains infrared (IR) spectroscopy as a vital tool for identifying molecular structures through absorption patterns. It details the quantum harmonic oscillator model relevant to diatomic molecules, such as hydrogen halides, and the calculation of bond stretching in HCl. The page also covers selection rules for IR transitions that require changes in dipole moment and discusses factors affecting molar absorptivity and transition probabilities.
- 4.7: Vibrational Overtones
- This page discusses the limitations of the harmonic oscillator model for molecular vibrations, particularly at higher energy levels where anharmonic effects become significant. The anharmonic oscillator model, incorporating higher-order terms, offers more accurate predictions and allows for transitions between various vibrational states, resulting in overtones. Observed frequencies align better with anharmonic models, especially for higher energy levels, leading to weaker intensity lines.
- 4.8: Rotations Accompany Vibrational Transitions
- This page explains the rovibrational spectra of diatomic gas molecules, detailing vibrational and rotational transitions influenced by quantum selection rules and bond length changes. It covers energy quantization, resulting in P- and R-branches, and discusses the rotational constant \(B\) and the Q-branch in spectroscopy.
- 4.9: Unequal Spacings in Vibration-Rotation Spectra
- This page discusses the differences between real and ideal rovibrational spectra, emphasizing the effects of rotational-vibrational coupling and centrifugal distortion on line spacing in R-branch and P-branch as energy varies. It notes how bond length influences vibrational states and the rotational constant, detailing how the spacing in R-branch decreases with increasing J values, while P-branch spacing increases as J decreases.
- 4.10: Electronic Spectra Contain Electronic, Vibrational, and Rotational Information
- This section discusses the characteristics of electronic spectra which can involve the simultaneous change of electronic, vibrational and rotational quantum states.
- 4.11: The Franck-Condon Principle
- This page explains the Franck-Condon Principle, detailing how electronic transitions in spectroscopy occur with minimal nuclear change. It highlights the significance of the Franck-Condon overlap integral, connecting transition probabilities to vibrational wavefunction overlaps.
- 4.12: Electronically Excited Molecules can Relax by a Number of Processes
- This page covers the mechanisms of laser operation, emphasizing molecular transitions between excited and ground states, influenced by fluorescence and phosphorescence. It explains the differences in lifetime and energetics between these processes, including vibrational relaxation, internal conversion, and intersystem crossing.
- 4.13: Raman Spectroscopy
- Raman spectroscopy is a powerful tool for determining chemical species. As with other spectroscopic techniques, Raman spectroscopy detects certain interactions of light with matter. In particular, this technique exploits the existence of Stokes and Anti-Stokes scattering to examine molecular structure.
- 4.14: The Dynamics of Transitions can be Modeled by Rate Equations
- This page discusses Einstein's three processes of atomic spectral line formation: spontaneous emission, stimulated emission, and absorption. It explains the behavior of atoms under radiation, focusing on the interplay of these processes, thermal equilibrium, and the influence of temperature on electron distribution.
- 4.15: Beer's Law
- Beer's law connects absorbance to the concentration of the absorbing species. In this section we derive Beer's law and consider some of its limitations.
- 4.16: A Two-Level System Cannot Achieve a Population Inversion
- This page explains lasing in two-level atomic systems, highlighting key conditions for laser operation: coherence, monochromatic output, collimation, and efficiency. Coherence relies on stimulated emission and requires a population inversion, which is difficult to achieve in two-level systems due to high energy demands. As a result, these systems are mostly limited to pulsed operation, making three-level systems more favorable for continuous lasing.
- 4.17: Population Inversion can be Achieved in a Three-Level System
- This page explains the concept of optical pumping in laser systems, focusing on the limitations of two-level systems and the advantages of three-level systems, like ruby lasers, which allow population inversion through a metastable state. It describes the conditions for achieving population inversion via the decay rates between energy levels and introduces four-level lasers, such as He-Ne and Nd:YAG, which offer better efficiency and continuous output.
- 4.18: What is Inside a Laser?
- This page explains the three main components of lasers: the gain medium for light emission, the pump source for energizing the medium, and the optical cavity for light amplification. Gain mediums can be solid, liquid, or gas, with examples like ruby in solid-state lasers and noble gases in gas lasers. Liquid dye lasers are noted for their tunability. The pump source excites the medium, while the optical cavity enhances amplification.
- 4.19: The Helium-Neon Laser
- This page discusses the He-Ne laser, the first continuous-wave laser developed by Ali Javan and his team. It emits predominantly red light and is stable in wavelength and intensity, making it ideal for applications like holography and laser pointers. While it was dominant until the 1990s, it has been eclipsed by more affordable semiconductor lasers. However, due to its durability and low production costs, the He-Ne laser continues to maintain popularity.
- 4.20: Modern Applications of Laser Spectroscopy
- This page discusses the diverse applications of laser light in studying atomic and molecular interactions, highlighting its properties like monochromaticity and coherence. It mentions uses in tracking chemical reactions, biological processes, and art conservation, as well as advancements in chemical kinetics and spectroscopy through pulsed and high-intensity lasers.
- 4.21: Broadening Mechanisms
- Spectrum lines are not infinitesimally narrow; they have a finite width. A graph of radiance or intensity per unit wavelength (or frequency) versus wavelength (or frequency) is the line profile. There are several causes of line broadening, some internal to the atom, others external, and each produces its characteristic profile. Analysis of the exact shape of a line profile may give us information about the physical conditions, such as temperature and pressure, in a stellar atmosphere.