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Related Concept Videos

IR Absorption Frequency: Delocalization01:04

IR Absorption Frequency: Delocalization

Electron delocalization refers to the distribution of electrons across multiple atoms within a molecule rather than being confined to a single atom or bond. This phenomenon is common in systems with conjugated bonds—structures where alternating single and double bonds allow π-electrons to move freely across the network. The movement of electrons stabilizes the molecule and can affect various chemical properties, including vibrational frequencies observed in IR spectroscopy.
In IR spectroscopy,...
Inductive Effects on Chemical Shift: Overview01:27

Inductive Effects on Chemical Shift: Overview

The protons in unsubstituted alkanes are strongly shielded with chemical shifts below 1.8 ppm. Methine, methylene, and methyl protons appear at approximately 1.7, 1.2 and 0.7 ppm, while the proton signal from methane appears at 0.23 ppm. An electronegative substituent, such as chlorine, withdraws the electron density from the protons, increasing their chemical shift. Progressive substitution of the hydrogens in methane by chlorine shifts the proton signals increasingly downfield, to 3.05 ppm in...
π Electron Effects on Chemical Shift: Overview01:27

π Electron Effects on Chemical Shift: Overview

An applied magnetic field causes loosely bound π-electrons in organic molecules to circulate, producing a local or induced diamagnetic field over a large spatial volume. As the molecules tumble in solution, the field generated by π-electrons in spherical substituents results in a zero net field. However, the net field generated by π-electrons in non-spherical substituents is not zero. The effect of this induced field depends on the orientation of the molecule with respect to B0, resulting in...
Atomic Nuclei: Nuclear Relaxation Processes01:23

Atomic Nuclei: Nuclear Relaxation Processes

In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis. This...
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.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations01:08

IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations

Identical bonds within a polyatomic group can stretch symmetrically (in-phase) or asymmetrically (out-of-phase). Similar to hydrogen bonding, these vibrations also influence the shape of the IR peak. Generally, asymmetric stretching frequencies are higher than symmetric stretching frequencies. For example, primary amines exhibit two distinct IR peaks between 3300–3500 cm−1 corresponding to the symmetric and asymmetric N-H stretching, while secondary amines exhibit a single stretching vibration...

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Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light
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Delocalization-localization transition due to anharmonicity.

David Hajnal1, Rolf Schilling

  • 1Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, Mainz, Germany.

Physical Review Letters
|October 15, 2008
PubMed
Summary
This summary is machine-generated.

Energy localization in the Fermi-Pasta-Ulam chain does not require multiple conserved quantities. A critical amplitude triggers a sharp transition between delocalization and localization, with unique time scales observed near this point.

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Area of Science:

  • Nonlinear dynamics
  • Condensed matter physics
  • Computational physics

Background:

  • The Fermi-Pasta-Ulam (FPU) chain is a fundamental model in nonlinear dynamics, often studied to understand energy transport and localization phenomena.
  • Previous research suggested that multiple conserved quantities might be necessary for energy localization in such systems.

Purpose of the Study:

  • To investigate the conditions required for energy localization in a reduced Fermi-Pasta-Ulam chain.
  • To identify and characterize the transition between delocalized and localized energy states.
  • To explore the implications of energy localization on energy transport.

Main Methods:

  • Analytical calculations were performed on a simplified FPU chain model.
  • Numerical simulations were employed to observe energy dynamics and localization.
  • The study focused on analyzing the role of conserved quantities and critical amplitudes.

Main Results:

  • Energy localization was demonstrated to occur without needing more than one conserved quantity.
  • A sharp delocalization-localization transition was identified at a critical amplitude (A_c).
  • Diverging time scales were observed as the system approached A_c from both above and below.
  • Above A_c, energy packets evolved towards discrete breathers, yet ballistic energy transport persisted.

Conclusions:

  • The findings challenge the necessity of multiple conserved quantities for energy localization in FPU chains.
  • The critical amplitude A_c represents a significant transition point in the system's energy dynamics.
  • Ballistic energy transport can coexist with localized energy states, indicating a complex interplay between these phenomena.