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Related Concept Videos

Spin–Spin Coupling Constant: Overview01:08

Spin–Spin Coupling Constant: Overview

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 have a...
Spin–Spin Coupling: Three-Bond Coupling (Vicinal Coupling)01:22

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

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...
Atomic Nuclei: Nuclear Spin State Overview01:03

Atomic Nuclei: Nuclear Spin State Overview

NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
Spin–Spin Coupling: One-Bond Coupling01:17

Spin–Spin Coupling: One-Bond Coupling

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,...
Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

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...
Continuous Charge Distributions01:17

Continuous Charge Distributions

Imagine a bucket of water. It contains many molecules, of the order of 1026 molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.
The electric charge can also be subjected to an analogical...

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Study of Protein Dynamics via Neutron Spin Echo Spectroscopy
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Published on: April 13, 2022

Microcanonical relation between continuous and discrete spin models.

Lapo Casetti1, Cesare Nardini, Rachele Nerattini

  • 1Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, and INFN, sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy. lapo.casetti@unifi.it

Physical Review Letters
|March 17, 2011
PubMed
Summary
This summary is machine-generated.

Researchers propose that O(n) models with ferromagnetic interactions share critical energy densities with Ising models (n=1) at phase transitions. This conjecture is supported by findings in long-range and nearest-neighbor interactions across dimensions.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • Continuous spin models and Ising models are fundamental in statistical mechanics.
  • Understanding phase transitions and critical phenomena is crucial in these models.
  • The microcanonical density of states provides insight into system thermodynamics.

Purpose of the Study:

  • To propose an approximate expression for the microcanonical density of states.
  • To conjecture a relationship between critical energy densities of O(n) models and Ising models.
  • To investigate the validity of this conjecture for various interaction ranges and dimensions.

Main Methods:

  • Relating stationary points of continuous spin models to Ising model configurations.
  • Developing an approximate expression for the microcanonical density of states.
  • Comparing critical energy densities through theoretical analysis and numerical simulations.

Main Results:

  • A novel approximate expression for the microcanonical density of states was derived.
  • The conjecture posits that O(n) models with ferromagnetic interactions have critical energy densities equal to the Ising model (n=1).
  • The conjecture is proven for long-range interactions and numerically supported for n=2, 3 in 3D.

Conclusions:

  • The study establishes a significant link between continuous spin models and Ising models regarding phase transitions.
  • The findings suggest a universal behavior in critical energy densities across different values of n.
  • Further numerical investigation is warranted, particularly for the XY model in 2D.