Comparative analysis of electric field influence on the quantum wells with different boundary conditions: II. Thermodynamic properties.
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It is shown that even for the temperatures smaller than [Formula: see text] the total dipole moment [Formula: see text] may become negative for the quite moderate [Formula: see text].Different asymptotic cases of, e.g., the small and large temperatures and low and high voltages are derived analytically and explained physically.Parallels are drawn to the similar properties of the 1D harmonic oscillator, and similarities and differences between them are discussed.
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Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
ABSTRACT
Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary conditions (BCs) at its edges in the perpendicular to the surfaces electric field [Formula: see text] are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity [Formula: see text] for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature T for the pure Neumann QW and their absence for other BCs are predicted and explained by the structure of the corresponding energy spectrum. Applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential. A remarkable feature of the Fermi grand canonical ensemble is, at any BC combination in zero fields, a salient maximum of [Formula: see text] observed on the T axis for one particle and its absence for any other number N of corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different N. It is proved that critical temperature [Formula: see text] of the Bose-Einstein (BE) condensation increases with the applied voltage for any number of particles and for any BC permutation except the ND case at small intensities [Formula: see text] what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than [Formula: see text] the total dipole moment [Formula: see text] may become negative for the quite moderate [Formula: see text]. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed with N growing. Different asymptotic cases of, e.g., the small and large temperatures and low and high voltages are derived analytically and explained physically. Parallels are drawn to the similar properties of the 1D harmonic oscillator, and similarities and differences between them are discussed. No MeSH data available. Related in: MedlinePlus |
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Mentions: As a final example, Fig.9 exhibits evolution of the heat capacity and chemical potential with the varying electric field and temperature for the pure Neumann QW. It is seen that for the small number of bosons, say, in panel (a), the applied voltage leads, at quite warm sample, to the increase of while at the small T, the width of the temperature-independent zero-capacity plateau increases with the field. These features were discussed before for the canonical ensemble. Increasing the number of the particles in the well leads to the suppression of the voltage dependence, as a transition from panel (a) to (b) with and (c) for vividly demonstrates. No any noticeable field dependence is seen there in the range . It is well known that for some potentials, such as, e.g., the 3D isotropic harmonic trap [14, 16, 18, 19, 34, 35], the heat capacity has a cusp-like peculiarity as it passes through the critical temperature while for the 1D quadratic potential it is a smooth function of T [16, 18, 35]. Fig.9 exemplifies that no any peculiarity is observed for the 1D hard-wall potential with Neumann surfaces and arbitrary applied electric fields. Our calculations confirm that the same is true for any other BCs. Finally, panel (d) shows that the chemical potential μ is a monotonically decreasing function of both the electric field and temperature T. It is seen that the growing temperature diminishes the voltage influence on the chemical potential. |
View Article: PubMed Central - PubMed
Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary conditions (BCs) at its edges in the perpendicular to the surfaces electric field [Formula: see text] are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity [Formula: see text] for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature T for the pure Neumann QW and their absence for other BCs are predicted and explained by the structure of the corresponding energy spectrum. Applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential. A remarkable feature of the Fermi grand canonical ensemble is, at any BC combination in zero fields, a salient maximum of [Formula: see text] observed on the T axis for one particle and its absence for any other number N of corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different N. It is proved that critical temperature [Formula: see text] of the Bose-Einstein (BE) condensation increases with the applied voltage for any number of particles and for any BC permutation except the ND case at small intensities [Formula: see text] what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than [Formula: see text] the total dipole moment [Formula: see text] may become negative for the quite moderate [Formula: see text]. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed with N growing. Different asymptotic cases of, e.g., the small and large temperatures and low and high voltages are derived analytically and explained physically. Parallels are drawn to the similar properties of the 1D harmonic oscillator, and similarities and differences between them are discussed.
No MeSH data available.