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¹H NMR: Long-Range Coupling01:27

¹H NMR: Long-Range Coupling

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The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene...
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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...
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Fermi Level Dynamics01:12

Fermi Level Dynamics

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The vacuum level denotes the energy threshold required for an electron to escape from a material surface. It is usually positioned above the conduction band of a semiconductor and acts as a benchmark for comparing electron energies within various materials.
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Spin–Spin Coupling: One-Bond Coupling01:17

Spin–Spin Coupling: One-Bond Coupling

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Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
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Spin–Spin Coupling Constant: Overview01:08

Spin–Spin Coupling Constant: Overview

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In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
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Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

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Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
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Setting Limits on Supersymmetry Using Simplified Models
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Phase Models Beyond Weak Coupling.

Dan Wilson1, Bard Ermentrout2

  • 1Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA.

Physical Review Letters
|November 9, 2019
PubMed
Summary
This summary is machine-generated.

Higher-order effects in coupled oscillators are captured using isostable reduction, improving upon standard phase reduction methods for complex systems like neural models and the Kuramoto-Sivashinsky equations.

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Area of Science:

  • Nonlinear dynamics
  • Theoretical neuroscience
  • Mathematical physics

Background:

  • Coupled oscillator networks are fundamental to many natural and engineered systems.
  • First-order phase reduction simplifies analysis but neglects higher-order effects.
  • These neglected effects can lead to inaccurate predictions of network behavior.

Purpose of the Study:

  • To develop a more accurate method for analyzing coupled oscillator networks.
  • To incorporate higher-order dynamics lost in traditional phase reduction.
  • To demonstrate the limitations of standard methods in specific complex systems.

Main Methods:

  • Application of the theory of isostable reduction.
  • Analysis of weakly coupled complex Ginzburg-Landau equations.
  • Modeling of conductance-based neural networks.
  • Derivation of the Kuramoto-Sivashinsky equations.

Main Results:

  • Isostable reduction successfully incorporates higher-order effects.
  • Demonstrated failure of standard phase reduction in specific bifurcation scenarios.
  • Accurate modeling of complex dynamics in neural and fluid systems.

Conclusions:

  • Isostable reduction offers a more comprehensive framework for coupled oscillator analysis.
  • Higher-order effects are crucial for understanding bifurcations in complex networks.
  • This method provides a more robust tool for studying phenomena where standard phase reduction fails.