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Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model since it pictured the atom as a miniature “solar system” with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton as...
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Orbitals are the areas outside of the atomic nucleus where electrons are most likely to reside. They are characterized by different energy levels, shapes, and three-dimensional orientations. The location of electrons is described most generally by a shell or principal energy level, then by a subshell within each shell, and finally, by individual orbitals found within the subshells.
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Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.
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In an atom, the negatively charged electrons are attracted to the positively charged nucleus. In a multielectron atom, electron-electron repulsions are also observed. The attractive and repulsive forces are dependent on the distance between the particles, as well as the sign and magnitude of the charges on the individual particles. When the charges on the particles are opposite, they attract each other. If both particles have the same charge, they repel each other.
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Electrons are negatively charged subatomic particles attracted to and orbit around the positively-charged nucleus of an atom. They reside in spaces associated with energy levels called shells and are further organized into subshells and orbitals within each shell.
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Consider a system comprising several point masses. The coordinates of the center of mass for this system can be expressed as the summation of the product of each mass and its position vector divided by the total mass:
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Model and Energy Bounds for a Two-Dimensional System of Electrons Localized in Concentric Rings.

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Researchers studied spinless electrons in circular rings, finding the lowest energy configuration by minimizing Coulomb interactions using simulated annealing. This provides approximations for quantum ground state energy in low-dimensional systems.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Nanoscience

Background:

  • Electrons confined to circular orbits interact via Coulomb potential.
  • Classical model simplifies to minimizing interaction energy for lowest energy state.

Purpose of the Study:

  • Determine equilibrium energy and configurations for interacting spinless electrons in 2D rings.
  • Provide semi-classical approximations for quantum ground state energy.

Main Methods:

  • Simulated annealing algorithm for numerical determination of equilibrium energy.
  • Semi-classical approach to model the quantum system.

Main Results:

  • Accurate numerical determination of equilibrium energy and electron configurations.
  • Reliable approximations for quantum ground state energy for various system sizes.

Conclusions:

  • The simulated annealing method effectively finds low-energy states for 2D electron systems in rings.
  • The model offers insights into low-dimensional systems relevant to nanoscience and nanomaterials.