PAULI FORCE
(June 2023)
(June 2023)
ABSTRACT
The Pauli force is a virtual quantum mechanical force that occurs at distances of the order of the de Broglie wavelength of the particles: a force that prevents the particles from approaching each other and acts against their densification, due to the Pauli principle of inhibition. The Pauli force observed in the quantum confinement phenomenon acts against the repulsive force between electrons. The Pauli force can also be observed as a measurable increase in pressure when measuring thermal pressure: the pressure of the particle cloud may be higher than would be expected classically from the temperature, particle density. The cloud pressure is less dependent on temperature than in the case of an ideal Fermi gas.
INTRODUCTION
The Pauli force is interpreted using an empirical potential well and results in a virtual force field - virtual in the sense that the magnetic potential A is a physical field, while the magnetic field H is a virtual, mathematical field (and it is known that it is the potentials that must be retarded, not the virtual fields) - but the magnitude of the virtual Pauli force can only be calculated indirectly, from the Coulomb force, as an increment of pressure. It is modelled with a steep potential well and in the model the potential has to be retarded, the potential build-up is at the speed of light. The direction of the force is such that it reduces the effect of the Pauli violation, and estimates of its magnitude exist based on the magnitude of the reduction.
The properties of the degenerate electron gas, the quantum confinement phenomenon and their relationship have been investigated for a two-dimensional electron gas. The research was carried out via an internet search, with the aim of extending the validity of the force due to quantum confinement, known in solid state physics as the Pauli force in the manuscript, to gases and charged plasma-like gases.
Early examples of this phenomenon were observed not only at the solid-state interface, in semiconductors, but also on the outer surface of helium, where two-dimensional electron gas was observed. Sommer, W. T. (16 March 1964), "Liquid Helium as a Barrier to Electrons". Physical Review Letters 12 (11), pp. 271-273, published by the American Physical Society. DOI:10.1103/physrevlett.12.271. ISSN 0031-9007.) reported electrons that can move through states on the surface of liquid helium. The formation of conducting surface channels due to the appearance of surface states has also been observed for other insulating materials. Such phenomena occur, for example, in topological insulators (https://en.wikipedia.org/wiki/Topological_insulator). Here, we consider them to be valid for gases: a phenomenon related to the Pauli force may also be a spherical lightning.
Degenerate electron gas (https://en.wikipedia.org/wiki/Potential_well, https://en.wikipedia.org/wiki/Quantum_well): "Degeneracy is caused by quantum mechanical effects: it occurs when the distance of the gas particles is small compared to the de Broglie wavelength of the particles. Particles with half spin, electrons satisfy Fermi-Dirac statistics. E.g. gas expansion causes the superfluidity of helium at low temperatures. In the interior of stars, high-pressure electrons that are stripped from fully ionised atoms can become degenerate as "electron gas".
In the degenerate electron gas, Pauli's principle causes the electrons to experience forces acting between them to resist further convergence - the condensation - (which has nothing to do with the electrical Coulomb repulsion acting between the electrons).
In the degenerate electron gas, Pauli's principle causes the electrons to experience forces acting between them to resist further convergence - the condensation - (which has nothing to do with the electrical Coulomb repulsion acting between the electrons).
A Pauli force phenomenon may also be a spherical filament. In contrast, in the degenerate state, electrons cannot stay at the same energy level at high density due to the Pauli exclusion principle, and are forced to occupy states with higher energy. Higher energies, in turn, imply higher and higher velocities. And high-speed electrons are like fast-flying gas molecules: they exert a force during collisions, which leads to an increase in pressure. So electrons will have extra pressure because the Pauli principle forces them to higher energy levels, called the pressure of degenerate electrons. The pressure of the degenerate electron gas is higher than what would be expected classically from the temperature, particle density." (Note: The Pauli force associated with the pressure increase of the degenerate electron gas can balance the Coulomb pressure on a spherical surface.)
The probability of the quantum confinement phenomenon (https://en.wikipedia.org/wiki/Potential_well) is higher when the particle mass is small, and therefore it is most easily developed for electrons. The electron energy levels are then not continuous but discrete, with finite density of states. The energy states allowed for electrons are composed of quasi-continuous enegia bands: the width of the bands, the density of states, and the forbidden bands separating the bands determine the gas properties. There are discrete jumps in the density of states of the two-dimensional electron gas. Many allowed quantum states are possible, but they do not coalesce into coherent energy bands
"The excess pressure may at times be enough for the stellar core to resist a certain amount of gravity, preventing further collapse of the star's interior. At the end of the life of stars with an initial mass of less than 8 solar masses (i.e. brown dwarfs, red dwarfs and white dwarfs below 1.4 solar masses), the pressure of the degenerate electron gas stops gravity from compressing the star, stabilising its state and size. If the white dwarf has sucked in material from a neighbouring star, causing it to grow to over 1.4 solar masses, or if the core left over from a Type II supernova explosion is larger than 1.4 solar masses, in these cases gravity overcomes the "Pauli pressure" and squeezes electrons into protons, fusing them into neutrons, creating the neutron star. The degenerate gas pressure effect does not disappear, however, because the neutrons also have a spin of half, i.e. they are subject to Pauli's principle. In neutron stars, the pressure of the degenerate "gas" of neutrons stops the contraction forced by gravity. However, if the neutron star has a mass greater than 3 solar masses, the pressure of the degenerate neutrons is overcome by gravity and the neutron star collapses into a black hole. By the time the pressure of a degenerate gas can be overcome by gravity, the particles of the degenerate gas have been forced to energy levels so high that their speed is very close to the speed of light."
Two-dimensional electron gas (https://en.wikipedia.org/wiki/Two-dimensional_electron_gas)
a phenomenon that also occurs in semiconductor devices. When electrons are allowed to move freely in two directions of space while quantum confinement prevails in the third direction, a layer with specific electrical transport properties is formed in the material.
The quantum states allowed for the electrons are separated by discrete energy levels in the confinement direction, and their motion occurs only in the plane of the two-dimensional electron gas.
A two-dimensional electron gas can be created if the conduction bandwidth at the interface is below the Fermi level due to the band bending at thermal equilibrium. In this region, the density of states corresponding to the conduction band would be valid, so the semiconducting material becomes conducting in this channel. However, due to the spatial confinement, a quantum confinement phenomenon also occurs, so only energy quantized states are allowed perpendicular to the surface. (Note: The pressure gradient of the degenerate electron gas perpendicular to the surface is in equilibrium with the Coulomb repulsion, called the Pauli force.) The two-dimensional electron gas domain is modelled by considering solutions that are extended in the in-plane directions and perpendicular to the plane of the electron gas domain. The actual slope, gradient, of the potential well is related to the Pauli force distance and density dependence.
The quantum states allowed for the electrons are separated by discrete energy levels in the confinement direction, and their motion occurs only in the plane of the two-dimensional electron gas.
A two-dimensional electron gas can be created if the conduction bandwidth at the interface is below the Fermi level due to the band bending at thermal equilibrium. In this region, the density of states corresponding to the conduction band would be valid, so the semiconducting material becomes conducting in this channel. However, due to the spatial confinement, a quantum confinement phenomenon also occurs, so only energy quantized states are allowed perpendicular to the surface. (Note: The pressure gradient of the degenerate electron gas perpendicular to the surface is in equilibrium with the Coulomb repulsion, called the Pauli force.) The two-dimensional electron gas domain is modelled by considering solutions that are extended in the in-plane directions and perpendicular to the plane of the electron gas domain. The actual slope, gradient, of the potential well is related to the Pauli force distance and density dependence.
In a high density electron cloud, the combined effect of the two quantum mechanical principles -Heisenberg and Pauli- tends to reduce the density. If the average distance of the particles decreases below Δx, the momentum uncertainty Δp increases. According to quantum statistics, the 6-dimensional phase space of coordinates and moments can be partitioned into quantum cells of size h3. According to the Heisenberg and Pauli principles, each quantum cell can contain at most two fermions (with opposite spin). If all quantum cells were filled and the particles were to be squeezed together even more, a pressure of quantum origin (quantum pressure) would appear in the system, which does not depend on temperature but on the density of the particles. Such a state of matter is called a degenerate state (http://www.phy.bme.hu/~szunyogh/Fiz_GMK_MSc/Electrons_in_femic.pdf). It can be shown that a dense electron gas becomes degenerate. The pressure of the degenerate electron gas is higher than would be expected classically from the temperature and particle density.
If the concentration of electrons is ne and the average volume per electron is 1/ne ≈ L3 , where L is the average distance between electrons. If the density is very high, the distance will be very small then the Fermi momentum is the Heisenberg principle:
Δp ≈ h/ 2π L = h/2π ne 1/3 = pF, corresponding to a Fermi energy of pF 2/2me . The electron gas becomes degenerate when the Fermi energy exceeds the thermal energy kT, where T is the temperature in Kelvin degrees (k = 1.380649 10 -23 J/K, https://hu.wikipedia.org/wiki/Boltzmann-%C3%A1lland%C3%B3), and the gas has λ = h/(3me kT) 1/2 de Broglie wavelength. (https://hu.wikipedia.org/wiki/Szabadelektron-modell, http://astro.u-szeged.hu/oktatas/asztrofizika/html/node14.html) So degeneracy occurs faster if the mass of the particles is small. This is why electrons are the easiest to degenerate, more easily than protons. Substituting the Fermi energy term into the pressure integral formula, after elementary integration, gives the equation of state of the non-relativistic degenerate electron gas. If the electrons are of relativistic energy, a completely similar expression is obtained, only the constant and the exponent are different. From the pressure integral, we can deduce the magnitude of the Pauli force that balances the Coulomb repulsive force. Ideally, the surface conductivity is zero, increasing with increasing temperature and energy in discrete steps of 2e2/h.
*Wolfgang Pauli: https://www.termvil.hu/archiv/tv2000/tv0005/pauli.html