14: Nuclear Magnetic Resonance Spectroscopy

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Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a magnetic field absorb and re-emit electromagnetic radiation. This energy is at a specific resonance frequency which depends on the strength of the magnetic field and the magnetic properties of the isotope of the atoms. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals, and non-crystalline materials through nuclear magnetic resonance spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).

14.1: Nuclei Have Intrinsic Spin Angular Momenta
This page discusses the concept of spin in fundamental particles, highlighting that electrons, protons, and neutrons have a spin quantum number of $ \frac{1}{2} $, marking them as fermions, while bosons like photons have integer spins. The significance of spin lies in its role in defining electronic states of atoms and molecules, with a complete understanding requiring principles of relativistic quantum mechanics.
14.2: Magnetic Moments Interact with Magnetic Fields
This page discusses Werner Heisenberg's uncertainty principle from the 1920s, which states that increasing precision in an electron's position decreases accuracy in its momentum, reflecting a fundamental measurement limit in quantum mechanics. It illustrates these concepts by comparing the uncertainties of position and momentum between a baseball and an electron, emphasizing the distinct quantum behaviors and insights that emerge from such comparisons.
14.3: Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz
This page discusses the Heisenberg Uncertainty Principle, established by Werner Heisenberg in the mid-1920s. It states that more precise measurement of an electron's position results in greater uncertainty in its momentum, due to the wave nature of matter. This principle illustrates fundamental limits in quantum measurements by mathematically expressing the relationship between uncertainties in position and momentum.
14.4: The Magnetic Field Acting upon Nuclei in Molecules Is Shielded
This page explains how nuclei in isolation react to magnetic fields, contrasting this with the behavior of nuclei within molecules, which are affected by the magnetic fields of surrounding electrons. This interaction leads to two phenomena: shielding, where the nucleus experiences a lower magnetic field, and deshielding, where it experiences a higher field.
14.5: Chemical Shifts Depend upon the Chemical Environment of the Nucleus
This page discusses the chemical shift in NMR, which reveals information about the local molecular structure by measuring the local magnetic field around a nucleus, affected by electron currents and quantified by the shielding constant. It operates on a delta scale and includes contributions from local (diamagnetic and paramagnetic) and molecular factors (magnetic susceptibility from neighboring groups).
14.6: Spin-Spin Coupling Can Lead to Multiplets in NMR Spectra
This page explains spin-spin coupling in NMR spectroscopy, detailing how proton signals split into sub-peaks based on magnetic interactions between non-equivalent nuclei, following the n + 1 rule. It covers coupling constants for protons on sp3 and sp2 carbons and introduces complex coupling scenarios with multiple interacting proton sets.
14.7: Spin-Spin Coupling Between Chemically Equivalent Protons is Not Observed
This page discusses the concept of equivalent hydrogens in NMR spectroscopy, explaining that they do not affect each other's signals due to their identical chemical environments, resulting in singlet signals. It details the Hamiltonian matrix for spin-$1/2$ particles, eigenstates, and transitions in NMR, while distinguishing between equivalent and non-equivalent nuclei.
14.8: The n+1 Rule Applies Only to First-Order Spectra
This page explains the (n+1) Rule in NMR spectroscopy, which predicts the peak multiplicity based on equivalent coupled nuclei. It states that a nucleus connected to n equivalent nuclei will display n+1 peaks. The page illustrates how different hydrogen nuclei such as $H_a$, $H_b$, and $H_c$ appear as various peak patterns (singlets, doublets, etc.) based on their coupling to adjacent atoms. Each set of equivalent nuclei is analyzed separately to ascertain their specific splitting patterns.
14.9: Second-Order Spectra Can Be Calculated Exactly Using the Variational Method
14.E: Nuclear Magnetic Resonance Spectroscopy (Exercises)
This page includes exercises on magnetic dipole moments, nuclear spin operators, and their relationships to angular momentum. It covers quantum mechanics principles for a two-spin system, including Hermitian operators, energy corrections, resonance frequencies, and orthogonality of spin states. The text derives equations for energy differences, highlighting the influence of spin states on energy levels.

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