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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.
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...
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Double Resonance Techniques: Overview01:12

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Double resonance techniques in Nuclear Magnetic Resonance (NMR) spectroscopy involve the simultaneous application of two different frequencies or radiofrequency pulses to manipulate and observe two distinct nuclear spins. One important application of double resonance is spin decoupling, which selectively suppresses coupling with one type of nucleus while observing the NMR signal from another nucleus, simplifying the spectrum and enhancing resolution.
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Characteristics of Series Resonant Circuit01:24

Characteristics of Series Resonant Circuit

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Series resonance occurs in a circuit containing inductive (L), capacitive (C), and resistive (R) elements connected sequentially. At the resonance frequency, the inductive and capacitive reactances are equal in magnitude but opposite in sign, effectively canceling each other. This causes the circuit's impedance is minimal, primarily determined by the resistance R. The resonant frequency of an RLC circuit is defined as:
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Linear Approximation in Frequency Domain01:26

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Design Example: Underdamped Parallel RLC Circuit01:17

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Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
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Within the field of electrical circuits, source-free RLC circuits present an intriguing domain. These circuits comprise a series arrangement of a resistor, inductor, and capacitor, operating independently of external energy sources. Their initiation hinges upon utilizing the initial energy stored within the capacitor and inductor to instigate their functionality. Their mathematical equation, a second-order differential equation, sets these circuits apart. This equation captures how the...
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Coupled-mode-theory framework for nonlinear resonators comprising graphene.

Thomas Christopoulos1, Odysseas Tsilipakos2, Nikolaos Grivas1

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A new framework models nonlinear resonant structures with conductive materials like graphene. It reveals graphene

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Area of Science:

  • Nonlinear optics
  • Condensed matter physics
  • Materials science

Background:

  • Nonlinear resonant structures are crucial for advanced optical applications.
  • Modeling these structures often simplifies material properties, neglecting conductivity and dispersion.
  • Graphene's unique electronic properties offer potential for novel nonlinear devices.

Purpose of the Study:

  • To develop a general framework for analyzing nonlinear resonant structures with dispersive bulk and sheet materials.
  • To incorporate nonlinear current terms for conductive sheet materials.
  • To model and investigate bistability in a graphene-based traveling-wave resonator.

Main Methods:

  • A hybrid framework combining perturbation theory and coupled-mode theory.
  • Introduction of a nonlinear current term to account for conductive sheet materials.
  • Application to a graphene traveling-wave resonator system with third-order nonlinearity.
  • Validation against full-wave nonlinear finite-element simulations.

Main Results:

  • Graphene's complex conductivity disrupts the balance of electric and magnetic energies on resonance.
  • The dispersive nature of conductive materials significantly impacts the nonlinear response.
  • Neglecting linear dispersion leads to an underestimation of stored energy in surface currents.
  • Excellent agreement between the proposed framework and finite-element simulations.

Conclusions:

  • The developed framework accurately models nonlinear resonant structures with conductive materials.
  • Graphene's properties enable very low characteristic power for bistability.
  • Highlights the importance of considering material dispersion and conductivity in nonlinear analysis.
  • Demonstrates graphene's significant potential for nonlinear optical applications.