Using the semi-classical (Thomas–Fermi) approximation, the thermodynamic properties of ideal Fermi gases in a harmonic potential in an n-dimensional space are studied under the generalized uncertainty principle (GUP). The mean particle number, internal energy, heat capacity and other thermodynamic variables of the Fermi system are calculated analytically. Then, analytical expressions of the mean particle number, internal energy, heat capacity, chemical potential, Fermi energy, ground state energy and amendments of the GUP are obtained at low temperatures. The influence of both the GUP and the harmonic potential on the thermodynamic properties of a copper-electron gas and other systems with higher electron densities are studied numerically at low temperatures. We find: (1) When the GUP is considered, the influence of the harmonic potential is very much larger, and the amendments produced by the GUP increase by eight to nine orders of magnitude compared to when no external potential is applied to the electron gas. (2) The larger the particle density, or the smaller the particle masses, the bigger the influence of the GUP. (3) The effect of the GUP increases with the increase in the spatial dimensions. (4) The amendments of the chemical potential, Fermi energy and ground state energy increase with an increase in temperature, while the heat capacity decreases. TF0 is the Fermi temperature of the ideal Fermi system in a harmonic potential. When the temperature is lower than a certain value (0.22 times TF0 for the copper-electron gas, and this value decreases with increasing electron density), the amendment to the internal energy is positive, however, the amendment decreases with increasing temperature. When the temperature increases to the value, the amendment is zero, and when the temperature is higher than the value, the amendment to the internal energy is negative and the absolute value of the amendment increases with increasing temperature. (5) When electron density is greater than or equal to 1037 m−3, the influence of the GUP becomes the dominant factor affecting the thermodynamic properties of the system.
We report a low loss photonic crystal slab waveguide formed by deforming the innermost holes on either side of a single line defect so that the circular air holes are changed to elliptical holes. The group velocity and group velocity dispersion of this waveguide depend strongly on the innermost elliptical air holes. We obtain the photonic bands and group index of guided modes in this kind of photonic crystal waveguide by guided mode expansion method and investigate the dependence of photonic bands and group index of guided modes in this photonic crystal waveguide on the parameter of the innermost elliptical air holes. For waveguides with the optimum innermost elliptical air holes, we achieved a wider single mode region below the light line, in which light can easily propagate without intrinsic loss. At the same time, this kind of waveguide has nearly constant group velocity and vanishing group velocity dispersion in a 3—5 nm bandwidth. These results will be applicable to design and fabricate photonic crystal slab waveguides with low group velocity, low dispersion and low loss characteristics.
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