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Frustration and Conflict: Avoidance-Avoidance, Double-Approach Avoidance01:14

Frustration and Conflict: Avoidance-Avoidance, Double-Approach Avoidance

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Avoidance-avoidance conflict refers to a psychological situation where a person must choose between two or more unpleasant alternatives. These conflicts are particularly stressful because neither option is desirable. This dilemma is often expressed in sayings like "caught between a rock and a hard place" or "between the devil and the deep blue sea." For instance, individuals who fear dental procedures may find themselves torn between enduring a painful toothache or facing the...
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NMR Spectroscopy: Spin–Spin Coupling01:08

NMR Spectroscopy: Spin–Spin Coupling

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The spin state of an NMR-active nucleus can have a slight effect on its immediate electronic environment. This effect propagates through the intervening bonds and affects the electronic environments of NMR-active nuclei up to three bonds away; occasionally, even farther. This phenomenon is called spin–spin coupling or J-coupling. Coupling interactions are mutual and result in small changes in the absorption frequencies of both nuclei involved. While nuclei of the same element are involved...
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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.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must...
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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.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...
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Spin–Spin Coupling: Three-Bond Coupling (Vicinal Coupling)01:22

Spin–Spin Coupling: Three-Bond Coupling (Vicinal Coupling)

1.5K
Vicinal or three-bond coupling is commonly observed between protons attached to adjacent carbons. Here, nuclear spin information is primarily transferred via electron spin interactions between adjacent C‑H bond orbitals. This generally favors the antiparallel arrangement of spins, so 3J values are usually positive.
The extent of coupling depends on the C‑C bond length, the two H‑C‑C angles, any electron-withdrawing substituents, and the dihedral angle between the involved orbitals. The...
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Related Experiment Video

Updated: Jan 29, 2026

Walk with Me Hybrid Virtual/In-Person Walking for Older Adults with Neurodegenerative Disease
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Self-avoiding walk, spin systems and renormalization.

Gordon Slade1

  • 1Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 15, 2019
PubMed
Summary
This summary is machine-generated.

This study explores mathematical physics models like the self-avoiding walk and φ⁴ model. A rigorous non-perturbative renormalization group method advances understanding of critical exponents.

Keywords:
critical exponentrenormalizationself-avoiding walkφ4 model

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Area of Science:

  • Mathematical physics
  • Statistical mechanics
  • Condensed matter theory

Background:

  • Self-avoiding walks and lattice spin systems (e.g., φ⁴ model) are crucial in mathematics and physics.
  • Significant mathematical challenges persist, especially concerning critical exponents.

Purpose of the Study:

  • To survey unsolved mathematical problems in these models.
  • To present recent advancements in their mathematical understanding.

Main Methods:

  • Rigorous non-perturbative renormalization group (RG) method.
  • Application of RG techniques to lattice models.

Main Results:

  • Progress in rigorously understanding critical phenomena.
  • New insights into the mathematical properties of self-avoiding walks and φ⁴ models.

Conclusions:

  • The non-perturbative RG method offers a powerful tool for tackling complex problems in mathematical physics.
  • Advances provide a foundation for future research into critical exponents and related phenomena.