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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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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.
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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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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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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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Three identical single-phase transformers can be configured to form a three-phase transformer connection, which involves high-voltage and low-voltage windings. The high-voltage windings are denoted by capital letters A-B-C, while the low-voltage windings are labeled with lowercase letters a-b-c, representing their respective phases. This notation helps distinguish between the high and low voltage sides of the transformer.
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Winding coupling phase for pseudo-spin-derived topological photonics.

Tianyuan Liu1,2, Min Qiu3,4,5, Wei Yan6,7

  • 1College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang, China.

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|December 24, 2025
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Summary
This summary is machine-generated.

This study introduces a new theoretical framework for designing pseudo-spin-topological phases using evanescent coupling in photonic crystals. It enables novel spin-valley Hall phases and edge states without breaking time-reversal symmetry.

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Area of Science:

  • Topological photonics
  • Condensed matter physics
  • Materials science

Background:

  • Topological phases typically rely on spin-orbit interaction or antiferromagnetism.
  • Designing pseudo-spin-derived topological phases requires flexible theoretical frameworks.

Purpose of the Study:

  • To propose a flexible theoretical framework for designing pseudo-spin-derived topological phases.
  • To explore novel topological phases and edge states in photonic crystals.
  • To demonstrate practical silicon-on-insulator designs for these phenomena.

Main Methods:

  • Developing a theoretical framework based on evanescent coupling between resonators.
  • Utilizing a quantized coupling winding number to characterize topology.
  • Designing photonic crystals with tailored coupling winding numbers.
  • Proposing silicon-on-insulator (SOI) device designs.

Main Results:

  • Evanescent coupling exhibits a $\pi_1(S^1)$ topology with a quantized coupling winding number.
  • Demonstrated designs for spin-valley Hall phase (SVHP), anomalous Hall phase, and anti-helical edge states.
  • Achieved SVHP in a non-antiferromagnetic system without time-reversal symmetry breaking.
  • Designed anti-helical edge states independently of next-nearest coupling.

Conclusions:

  • The proposed framework offers a versatile and simple approach to designing pseudo-spin-topological phases.
  • The results are compatible with conventional fabrication processes.
  • Potential applications include spin-valley protected light transport and slow light guiding.