The New Theory of Electron Gas in a Magnetic Field and Tasks for Theory and Experiment

I. M. Dubrovskyi

G.V. Kurdyumov Institute for Metal Physics, NAS of Ukraine, 36 Academician Vernadsky Blvd., UA-03142 Kyiv, Ukraine

Received: 22.09.2015. Download: PDF

As shown, the conventional theory for an electron gas of metal in a magnetic field (based on Landau’s work) leads to some contradictions in both the theory and the comparison to experiments. The contradictions, which cannot be eliminated in the Landau theory, are considered. The magnetic moment of gas equals to zero in the conventional theory just as in a classical statistical mechanics as well as in a quantum statistical mechanics, when the summation over the occupied states is replaced by the integration. However, a classical charged particle in a magnetic field moves in its circular orbit and creates the diamagnetic moment. In the quantum theory, an eigenstate of particle in a magnetic field has always the average diamagnetic moment. The sum of equally directed magnetic moments cannot be equal to zero. The amendment to replacing summation by integration is taken into account in the Landau theory. Then, the magnetic moment is proved proportional to the volume, and the specific magnetic susceptibility of an electron gas is obtained. The Schrödinger equation of a system in a magnetic field has the complex coefficients. Therefore, an eigenstate has the nonzero field of average current density, all lines of which are closed in the considered area. The sum of these fields over all occupied states generates a non-uniform magnetization. Therefore, the specific magnetic susceptibility is meaningless. The energy spectrum of 2D-electron in a magnetic field in the Landau theory is equidistant peaks (Landau levels). The degeneracy multiplicities of these peaks are equal in magnitude. This spectrum is in contradiction with the mathematical theorems about the eigenvalues of Schrödinger equation with zero boundary condition. In the conventional theory, the homogeneous filling of whole area by the electron gas in a magnetic field is assumed. A crystal lattice, which is charged positively, is replaced by homogeneous background. Only under these assumptions, the average electric field that exerts on an electron is zero. However, as known, a magnetic field hinders homogeneously spreading of the electron gas in the plane perpendicular to it. These contradictions are eliminated in the new theory. The review of its principles and results obtained so far is presented in this paper. A. Ya. Khinchin has suggested the radically new method of theoretical justification of statistical mechanics. In the case under study, it requires that the space of system feasible states has to be defined not only by the energy eigenvalue, but also by the eigenvalue of angular momentum relative to axis that is parallel to the magnetic field. The Hamiltonian of electron gas in magnetic field contains energy of interaction between the total angular momentum and the magnetic field. This term is commutative with Hamiltonian gas, and its eigenvalue has to be equal to zero. Therefore, it can be eliminated. Then, the gas Hamiltonian will be identical with those of gas of interacting electrons in potential field. This problem is considered by means of the density functional method. As shown, the Landau levels are a result of mathematical mistake.

**Keywords:**
electron gas, magnetic field, angular momentum, statistical operator, magnetic moment, density of gas, density of states.

**PACS:**
05.20.Gg, 05.30.Ch, 05.30.Fk, 51.60.+a, 71.10.Ca, 71.70.Di, 75.20.-g

**DOI:**
https://doi.org/10.15407/ufm.17.01.053

**Citation:**
I. M. Dubrovskyi, The New Theory of Electron Gas in a Magnetic Field and Tasks for Theory and Experiment, *Usp. Fiz. Met.*, **17**, No. 1: 53—81 (2016) (in Russian), doi: 10.15407/ufm.17.01.053

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