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Related Concept Videos

¹H NMR: Long-Range Coupling01:27

¹H NMR: Long-Range Coupling

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 π orbitals.
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

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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Phase Transitions

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Phase space gaps and ergodicity breaking in systems with long-range interactions.

Freddy Bouchet1, Thierry Dauxois, David Mukamel

  • 1Laboratoire de Physique, Université de Lyon, CNRS, and Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon cedex 07, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 21, 2008
PubMed
Summary

This study on a generalized XY model reveals ergodicity breaking in both ferromagnetic and paramagnetic phases. This leads to unusual stable phases, like ferromagnetism within disorder.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • Mean-field spin models are crucial for understanding magnetic phenomena.
  • Ergodicity breaking, a deviation from typical system behavior, has been observed in various spin models.

Purpose of the Study:

  • To investigate a generalized isotropic XY model with two- and four-spin interactions.
  • To explore the occurrence and implications of ergodicity breaking in this model.
  • To map the microcanonical phase diagram and identify phase transitions.

Main Methods:

  • Exact solution in the microcanonical ensemble.
  • Calculation of the model's entropy.
  • Derivation of the microcanonical phase diagram.

Main Results:

  • Identified parameter regions exhibiting gaps in magnetization at fixed energy, indicating ergodicity breaking.
  • Calculated entropy and derived the microcanonical phase diagram.
  • Observed first-order phase transitions between ferromagnetic and paramagnetic phases.
  • Found ergodicity breaking occurring in both ferromagnetic and paramagnetic phases.

Conclusions:

  • The generalized XY model demonstrates ergodicity breaking, leading to unusual phase coexistence.
  • Ergodicity breaking challenges conventional phase transition understanding in magnetic systems.
  • The model provides insights into complex magnetic behaviors and phase stability.