[Paper Review] Relativistic Thermodynamics of Magnetized Fermi Electron Gas

Н. Л. Цинцадзе, Levan N. Tsintsadze|arXiv (Cornell University)|Dec 12, 2012

Cold Atom Physics and Bose-Einstein Condensates2 references3 citations

TL;DR

This paper develops a relativistic thermodynamic framework for a magnetized Fermi electron gas by deriving the relativistic thermodynamic potential using the relativistic Fermi distribution with quantized electron motion in a strong magnetic field. It reveals a novel adiabatic magnetization process that cools the electron gas to ultra-low temperatures, introduces a new quantum magnetosound velocity, and demonstrates oscillatory thermodynamic responses due to Landau quantization and relativistic effects.

ABSTRACT

To study the relativistic thermodynamic properties of a Fermi gas in a strong magnetic field, we construct the relativistic thermodynamic potential by the relativistic Fermi distribution function taking into account that the motion of particles in a plane perpendicular to the magnetic field is quantized. With this general potential at hand, we investigate all the thermodynamic quantities as a function of densities, temperatures and the magnetic field. We obtain a novel set of adiabatic equations. Having the expression of the pressure and adiabatic state equations, we determine the sound velocity for several cases revealing a new type of sound velocity. Finally, we disclose the magnetic cooling in the quantized electron Fermi gas, which is based on an adiabatic magnetization in contrast to the known adiabatic demagnetization.

Motivation & Objective

• To develop a relativistic thermodynamic description of a degenerate electron gas under strong magnetic fields, accounting for Landau quantization and relativistic effects.
• To investigate how the equation of state, entropy, specific heat, and sound velocity are modified by strong magnetic fields in both nonrelativistic and ultra-relativistic regimes.
• To explore the possibility of magnetic cooling via adiabatic magnetization as an alternative to conventional adiabatic demagnetization.
• To derive and analyze the quantum magnetosound velocity, identifying a new type of sound propagation influenced by magnetic field quantization.
• To examine the emergence of diamagnetism and superconducting transitions in strongly magnetized relativistic Fermi gases.

Proposed method

• Derives the relativistic thermodynamic potential using the relativistic Fermi distribution function with quantized transverse electron motion in a magnetic field.
• Applies Landau quantization to model the energy spectrum of electrons in a magnetic field, with discrete Landau levels: $ E_{\perp} = (\ell + \frac{1}{2})\hbar\omega_c $.
• Uses the grand canonical ensemble to compute thermodynamic quantities such as pressure, entropy, and specific heat from the thermodynamic potential.
• Applies low-temperature expansions (e.g., $ \sinh x \approx x $) to extract oscillatory and smooth contributions to thermodynamic functions.
• Derives adiabatic equations by enforcing constant entropy conditions, leading to temperature dependence on magnetic field strength.
• Calculates sound velocity from the derivative of pressure with respect to density at constant entropy, revealing a new quantum magnetosound mode.

Experimental results

Research questions

• RQ1 How does the relativistic thermodynamic potential of a magnetized Fermi electron gas depend on magnetic field, temperature, and electron density?
• RQ2 What are the effects of Landau quantization and relativistic corrections on the entropy and specific heat of the electron gas?
• RQ3 Can adiabatic magnetization lead to cooling in a relativistic Fermi electron gas, and how does it compare to adiabatic demagnetization?
• RQ4 What is the form of the sound velocity in a strongly magnetized relativistic electron gas, and does it exhibit a new quantum magnetosound mode?
• RQ5 How do oscillatory contributions to the magnetic susceptibility and thermodynamic functions arise from the interplay of quantum and relativistic effects?

Key findings

• The thermodynamic potential exhibits oscillatory behavior due to Landau quantization, with oscillations in susceptibility, entropy, and specific heat at high magnetic fields.
• The entropy and specific heat show oscillatory dependence on the magnetic field, with the oscillation frequency proportional to $ \frac{m_e c^2 (\overline{\mu}^2 - 1)}{\hbar \omega_c} $, and the oscillatory part dominates when $ \hbar \omega_c \ll m_e c^2 (\overline{\mu}^2 - 1) $.
• The adiabatic magnetization process leads to cooling with $ T \sim 1/H $ in the ultra-relativistic limit and $ T \sim 1/H^2 $ in the nonrelativistic limit, offering a novel cooling mechanism.
• The sound velocity in the nonrelativistic limit includes a quantum magnetosound contribution: $ C_S^2 = \frac{v_F^2}{3} + \frac{(\hbar\omega_c)^{3/2}}{2\pi(3\pi^2 n_e)^{1/3}\hbar m_e^{1/2}} \sin\left(\frac{\pi p_F^2}{m_e \hbar \omega_c} - \frac{\pi}{4}\right) $, indicating a new type of sound wave.
• The system exhibits strong diamagnetism due to Landau quantization, and the transition to a superconducting state is predicted in strong magnetic fields.
• The oscillatory thermodynamic response is most prominent when $ \frac{m_e c^2 (\overline{\mu}^2 - 1)}{\hbar \omega_c} \gg 1 $, confirming the quantum nature of the magnetic response.