Research On The Piezoelectric And Dielectric Storage Properties Of Lead-free Piezoelectric Ceramics

In this paper, analytical expressions for the thermodynamical properties of a two-dimensional electron gas in a perpendicular magnetic field are derived. This isaccomplished by first deriving the general expression for the thermodynamical potential,and then employing this result to obtain the corresponding expression for the two-dimensional gas. The chemical potential and magnetization are studied as a function oftemperature and magnetic field, and shown to be in agreement with prior work. It is alsoshown that the results are close to those obtained by assuming a Gaussian density of statesfor the Landau levels.using the general expression for the thermodynamical potential,then we study the specificheat of a noninteracting electron gas confined in one dimension by a harmonic-oscillatorpotential as a model of a metallic film with specular surfaces. The exact analyticalexpressions for the thermodynamical potential are obtained by using a formula of contourintegration, the specific heat,magnetization, magnetic susceptibility then is studied as afunction of the strength of the magnetic field at different regimes of the temperature andeffective thickness. The temperature and size effects on the behavior of the oscillatory partas a function of the strength of the magnetic field are also discussed，and some meaningfulresults are obtained.