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Mathematics

An Analytical Solution to the Problem of Hydrogen Isotope Passage through Composite Membranes Made from 2D Materials

Alexey Mikhailovich Bubenchikov, Mikhail Alekseevich Bubenchikov, Anna Sergeevna Chelnokova, Soninbayar Jambaa

Published Sep 22, 2021• 0 citations
Publisher

Abstract

An analytical solution to the problem of wave transport of matter through composite hyper-fine barriers is constructed. It is shown that, for a composite membrane consisting of two identical ultra-thin layers, there are always distances between the layers at which the resonant passage of one of the components is realized. Resonance makes it possible to separate de Broiler waves of particles with the same properties, which differ only in masses. Broad bands of hyper-selective separation of a hydrogen isotope mixture are found at the temperature of 40 K.

DOI
10.3390/math9192353

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An analytical solution to the problem of wave transport of matter through composite hyper-fine barriers is constructed. It is shown that, for a composite membrane consisting of two identical ultra-thin layers, there are always distances between the layers at which the resonant passage of one of the components is realized. Resonance makes it possible to separate de Broiler waves of particles with the same properties, which differ only in masses. Broad bands of hyper-selective separation of a hydrogen isotope mixture are found at the temperature of 40 K.
Abstract The use of hydrogen and helium as a fuel requires well-developed technologies for the separation of gas mixtures, in particular, separation of isotopes of an element. The presence of quantum effects makes such processes possible, and the proposed system of two potential barriers enhances the differences in the passage of isotopes through two layers of matter providing high separation efficiency. The probability of atoms passing through composite barriers is derived from the solving of stationary Schrodinger equation.
In this paper, the question about the use of wave dynamics for solving problems of membrane separation of helium isotopes in the gas state at cryogenic temperatures is considered. The dimensionless form of the stationary Schrödinger differential equation is obtained. Following that, the integral representation form of the wave function is written. This form, which is equivalent to the classical equation, is similar to the integral equation with a degenerate core; however, it contains a modulus of the argument with a shift along the real axis. Using the shift operator, the existing exponential function in the Schrödinger integral equation can be split into a differential operator and an exponential function of the argument module which does not contain a shift. The Fourier identity allows reducing the exponent of the modulus of the argument to a regular exponent. Next, based on the general property of a differential operator acting on an exponent, it is possible to calculate the spectral functions of the problem and write down the distribution for the wave function. This distribution is ultimately expressed through the spectra of the potential barrier. Thereafter, the structure and the spectrum of the composite barrier are considered. With the expression determining the reflection coefficient, it is found that the double-barrier system can have a resonant passage of one of the components in the sequence of distances between the layers of the membrane.
Wave-corpuscle duality is a property of nature whereby microscopic material objects can exhibit the properties of classical waves under certain conditions and the properties of classical particles under others. This property is typically demonstrated by observing interference as particles penetrate a barrier consisting of two slits. Below, we consider how the wave properties of a corpuscle in the form of an atom can influence a nearby identical atom, which exhibits the same wave properties. Atoms arranged in this manner represent coupled, identical oscillators, between which energy can be transferred to the maximum extent possible from one oscillator to the other. As a result, a nuclear reaction is possible, in which some nucleons from the nucleus of one atom are transferred to the nucleus of another atom. A list of all such reactions between all naturally occurring nuclides is provided, and it is shown that such reactions are responsible for the difference in the elemental composition of the Earth's crust from the elemental composition of the interstellar matter from which the Earth's crust formed. This confirms the presence of wave properties in atoms, leading to nuclear reactions between identical atoms.
The famous 1927 Solvay Conference was considered a turning point in the world of Physics. The scientific realists like Albert Einstein had lost and the instrumentalists like Niels Bohr had won the fundamental conflict. Since then Physics has followed the path of the instrumentalists in which Quantum Physics has been determined by the concept of Elementary Particles and Probability Waves. Until May 2013, when for the first time a photo had been made of the Probability Waves Inside the Hydrogen Atom . It should be impossible to make a photo of a non-existing complex probability wave with mass zero But the fact could not be denied anymore that scientists had succeeded to make the complex and non-existing visible. This will be the fundamental question in this article. Does an alternative theory exist in which the existence of this particular photo could be explained. And that the famous photo of the Hydrogen Atom can be explained in a logical and a simple way. To find the answer, we have to go back in time. Back 300 years in time to the moment when Isaac Newton discovered the beauty and one of the many secrets of the light. The moment when science walked away from light. And we return to the fundamental question: “Is it possible to build matter out of Light (confined electromagnetic waves)?” To answer this question we first have to answer the question how we have come to this world of Probability Waves and Elementary Particles and Quantum Physics.Louis de Broglie described in 1924 in his PhD thesis the wave properties of matter. Erwin Schrödinger published in 1926 the well-known Schrödinger wave equation with the characteristic spherical and elliptical wave solutions. To describe these material waves, the immediately and first explanation was: “There is only one possibility. These are “electromagnetic waves”. De Broglie waves are ” Light Waves”. Because these were the only waves, known in the world of Physics. But there was a fundamental problem. These material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger could never be solutions of the well-known 4 equations of James Clerk Maxwell. The material waves were spherical and elliptical solutions and that was impossible for the solutions of the Maxwell Equations. According to the Maxwell Equations, light always travels along a straight path. A second problem was that the model of the atom of Bohr could not hold. How could an electron be everywhere at the same time in a kind of a spherical shell surrounding the nucleus. And be a particle at the same time. And Bohr found a compromise to declare the “De Broglie” waves to be the probability waves. Two problems were solved. Bohr’s planetary model could hold. And there was a name for the unknown material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger. The theory of “Probability Waves, Elementary Particles and Quantum Physics” had been created. To find a possibility, to create matter out of light we have to go back to the Equations of James Clerk Maxwell. The only possibility to declare the probability waves, discovered by Louis de Broglie, to be electromagnetic waves, to be light, and to be the building element of our universe, is to find evidence that he Maxwell Equations are not complete and that it is possible to confine light (electromagnetic waves) in spherical and elliptical shapes. A new equation has to be found. In this article it has been claimed that this missing equation is equation (5). Because the Theory of General Relativity has been built on the Einstein-Maxwell Equation, a new equation has to be found that describes in a generalized way the interaction between light and matter. This missing equation is equation (5-a).