A report presents a theoretical study of the thermodynamics of an ultralow-temperature gas of fermions that interact with a gravitational field and with an externally imposed trapping potential but not with each other. The gravitational field is taken to define the z axis and the trapping potential to be of the form (m/2) (ωxx2+ωyy2+ωzz2), where m is the mass of a fermion; x, y, and z are Cartesian coordinates originating at the center of the trap; and the ω values denote effective harmonic- oscillator angular frequencies with respect to motion along the respective coordinate axes. The single-particle energy is found from the solution of the time-dependent Schroedinger equation for a Hamiltonian that includes kinetic energy plus the gravitational and trapping potentials. The equation for the single-particle energy is combined with Fermi statistics to obtain equations for the chemical potential, internal energy, and specific heat of the gas; the number of trapped fermions; and the spatial distribution of fermions at zero temperature. The equations reveal the ways in which the Fermi energy, the specific heat, and the shape of the Fermion cloud are affected by the gravitational field and the anisotropy of the trapping field.
This work was done by Igor Kulikov of Caltech for NASA’s Jet Propulsion Laboratory. To obtain a copy of the report, “An Influence of Gravitational Field on Properties of Trapped Fermions,” access the Technical Support Package (TSP) free on-line at www.nasatech. com/tsp under the Physical Sciences category. NPO-30248
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Effect of Gravitation on Noninteracting Trapped Fermions
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Overview
The document presents a theoretical study on the thermodynamics of a noninteracting ultralow-temperature gas of fermions subjected to an external gravitational field and a trapping potential. Conducted by Igor K. Kulikov at the California Institute of Technology for NASA's Jet Propulsion Laboratory, the research focuses on how these external influences affect the properties of the fermionic gas.
The trapping potential is modeled as a harmonic oscillator, defined by the equation ((m/2)(\omega_x x^2 + \omega_y y^2 + \omega_z z^2)), where (m) is the mass of a fermion, and (\omega_x), (\omega_y), and (\omega_z) are the angular frequencies corresponding to the x, y, and z axes. The study employs the time-dependent Schrödinger equation to derive the single-particle energy levels, which are then integrated with Fermi statistics to formulate equations for key thermodynamic quantities, including the chemical potential, internal energy, specific heat, and the number of trapped fermions.
The findings reveal significant insights into how the gravitational field and the anisotropic nature of the trapping potential influence the Fermi energy, specific heat, and the spatial distribution of fermions at zero temperature. The research highlights the importance of these factors in understanding the behavior of fermionic gases in microgravity conditions, which is particularly relevant for experiments conducted in space.
Additionally, the document emphasizes the novelty of this work, noting that it provides new knowledge regarding the thermodynamics of fermions at ultra-low temperatures in magnetic traps, specifically under the influence of non-homogeneous gravitational fields. The results are expected to be beneficial for future experiments involving trapped fermions, particularly in the context of measuring gravity gradients.
Overall, this study contributes to the broader understanding of quantum gases and their behavior in varying gravitational environments, paving the way for advancements in both fundamental physics and practical applications in space exploration and technology. The research underscores the interplay between quantum mechanics and gravitational effects, offering a unique perspective on the properties of ultracold fermionic systems.
