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The Bohr–Sommerfeld model (also known as the Sommerfeld model or Bohr–Sommerfeld theory) was an extension of the Bohr model to allow elliptical orbits of electrons around an atomic nucleus. Bohr–Sommerfeld theory is named after Danish physicist Niels Bohr and German physicist Arnold Sommerfeld.
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Nov 16, 2023 · Expanding on the works of Paul Drude and Hendrik Lorentz, Sommerfeld introduced the theory of free electrons in metals, advancing our understanding of electrical conductivity in solids. His model, known as Sommerfeld's theory, had a significant impact on solid-state physics and electronics.
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Arnold Johannes Wilhelm Sommerfeld, ForMemRS [1] (German: [ˈzɔmɐˌfɛlt]; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretical physics.
- Ideas and Assumptions
- From The Drude Model
- Properties of An Electron Gas
- Corrections to Drude's Model
- Inaccuracies and Extensions
- See Also
In the free electron model four main assumptions are taken into account:[Ashcroft & Mermin 2] 1. Free electron approximation: The interaction between the ions and the valence electrons is mostly neglected, except in boundary conditions. The ions only keep the charge neutrality in the metal. Unlike in the Drude model, the ions are not necessarily th...
Many physical properties follow directly from the Drude model, as some equations do not depend on the statistical distribution of the particles. Taking the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only changes the results related to the speed of the electrons.[Ashcroft & Mermin 3] Mainly, the free ...
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the Fermi energy as[Ashcroft & Mermin 5] 1. E F = ℏ 2 2 m e ( 3 π 2 n ) 2 3 , {\displaystyle E_{\rm {F}}={...
Heat capacity
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the heat capacity of metals. While most solids had a constant volumetric heat capacity given by Dulong–Petit law of about 3 n k B {\displaystyle 3nk_{\rm {B}}} at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity. The classical calculation using Drude'...
Mean free path
Notice that without the relaxation time approximation, there is no reason for the electrons to deflect their motion, as there are no interactions, thus the mean free path should be infinite. The Drude model considered the mean free path of electrons to be close to the distance between ions in the material, implying the earlier conclusion that the diffusive motion of the electrons was due to collisions with the ions. The mean free paths in the free electron model are instead given by λ = v F τ...
Thermal conductivity and thermopower
While Drude's model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities. The thermal conductivity is given by κ = c V τ ⟨ v 2 ⟩ / 3 {\displaystyle \kappa =c_{V}\tau \langle v^{2}\rangle /3} for free particles, which is proportional to the heat capacity and the mean free path which depend on the model (⟨ v 2 ⟩ 1 / 2 {\displaystyle \langle v^{2}\rangle ^{1/2}} is the mean (square) speed of the electrons...
The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:[Ashcroft & Mermin 13] Temperature dependence 1. The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. T...
Arnold Sommerfeld is one of the pioneers of quantum and atomic physics. He was a brilliant German theoretical physicist who also educated many students in physics. He is known for introducing the azimuthal quantum number (2nd quantum number) and the spin quantum number (4th quantum number).
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Mar 12, 2024 · Niels Bohr's model, which depicted electrons orbiting the nucleus like planets around the sun, was awarded a Nobel Prize in 1922 despite being technically incorrect. Arnold Sommerfeld enhanced the Bohr model in 1916 with elliptical orbits.
