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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.
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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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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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Ionic Association

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The ionic association is the association of oppositely charged ions in an electrolyte solution to form ion pairs. Bjerrum defined ion pairs as two oppositely charged ions whose electrostatic attraction exceeds the thermal energy of the system, typically expressed as 2kT. Electrostatic attraction depends on ionic charge, separation distance, and the dielectric constant of the medium. Thermal energy, represented by kT, reflects the tendency of ions to move independently due to molecular motion.
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Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
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Disordered Kitaev chains with long-range pairing.

Xiaoming Cai1

  • 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|January 19, 2017
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Summary

We investigated how disorder and superconductivity compete in a one-dimensional Kitaev model. A smaller exponent α in the superconducting pairing power law facilitates an earlier transition to the Anderson localized phase.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Materials
  • Topological Superconductivity

Background:

  • The generalized Kitaev model describes one-dimensional spinless fermions with long-range p-wave superconducting pairing.
  • Superconducting pairing decays with distance as a power law ~l^-α.
  • Understanding the interplay between disorder and superconductivity is crucial for novel quantum phenomena.

Purpose of the Study:

  • To investigate the competition between disorder and superconductivity in a generalized Kitaev model with incommensurate potentials.
  • To analyze the transition from a topological superconducting phase to a topologically trivial Anderson localized phase.
  • To determine the effect of the exponent α on this phase transition.

Main Methods:

  • Numerical determination of phase transition points.
  • Analysis of zero-mode Majorana fermion decay in the topological phase.
  • Study of single-particle states and correlation functions in the Anderson localized phase.

Main Results:

  • In the topological superconducting phase, Majorana fermion amplitude exhibits hybrid exponential-algebraic decay.
  • In the Anderson localized phase, critical single-particle states increase with decreasing α.
  • Correlation functions show short-range exponential and long-range algebraic decay, except at critical disorders.
  • Smaller α values lead to earlier topological phase transitions at lower disorder strengths.

Conclusions:

  • The exponent α significantly influences the competition between disorder and superconductivity.
  • The system transitions from a topological superconducting state to an Anderson localized state as disorder increases.
  • The decay exponent α dictates the critical disorder strength and the nature of localized states.