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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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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.
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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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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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Energy Diffusion in the Long-Range Interacting Spin Systems.

Hideaki Nishikawa1,2, Keiji Saito1

  • 1Kyoto University, Department of Physics, Kyoto 606-8502, Japan.

Physical Review Letters
|October 19, 2025
PubMed
Summary
This summary is machine-generated.

Energy diffusion in spin systems with long-range interactions is explored. Normal diffusion occurs when the interaction decay exponent α exceeds 3/2 in 1D and D in D≥2 dimensions.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Quantum Spin Systems

Background:

  • Investigating energy diffusion in spin systems with long-range interactions (V(r)∝r^{-α}) is crucial for understanding thermal transport.
  • The thermodynamic extensivity requires the exponent α to be greater than the lattice dimension D (α>D).

Purpose of the Study:

  • To determine the conditions for normal and anomalous energy diffusion in D-dimensional spin systems with algebraic long-range interactions.
  • To establish a general criterion for normal diffusion in one-dimensional systems.

Main Methods:

  • Analysis of the transverse Ising and XYZ spin models in D dimensions.
  • Application of fluctuating hydrodynamics and mathematical proofs for joint cumulants and current correlations.

Main Results:

  • In 1D, both normal and anomalous diffusion are observed, with anomalous diffusion linked to enhanced equilibrium current correlations.
  • A general theorem proves power-law clustering for joint cumulants.
  • The sufficient condition for normal diffusion in 1D is α>3/2, regardless of the specific spin model.
  • Lévy diffusion for α<3/2 is explained by fluctuating hydrodynamics, confirming the optimality of the condition.
  • In D≥2 dimensions, normal diffusion is observed for α>D.

Conclusions:

  • The critical exponent α dictates the diffusion behavior in long-range interacting spin systems.
  • A unified understanding of energy diffusion across different dimensions and interaction ranges is achieved.
  • The findings provide a clear criterion for distinguishing between normal and anomalous diffusion in these systems.