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Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties
Authors:O. Olendski
Affiliation:Department of Applied Physics and Astronomy, University of Sharjah, Sharjah, United Arab Emirates
Abstract:A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length Λ in the electric field urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0001 that pushes the particle to the surface is presented both in the canonical and two grand canonical representations and in the whole range of the Robin distance with the emphasis on its negative values which for the voltage‐free configuration support negative‐energy bound state. For the canonical ensemble, the heat capacity at urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0002 exhibits a nonmonotonic behavior as a function of the temperature T with its pronounced maximum unrestrictedly increasing for the decreasing fields as urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0003 and its location being proportional to urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0004. For the Fermi‐Dirac distribution, the specific heat per particle urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0005 is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the T axis by the plateau whose magnitude at the vanishing urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0006 is defined as urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0007, with N being a number of the particles. The maximum of urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0008 is the largest for urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0009 and, similar to the canonical ensemble, grows to infinity as the field goes to zero. For the Bose‐Einstein ensemble, a formation of the sharp asymmetric feature on the urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0010T dependence with the increase of N is shown to be more prominent at the lower voltages. This cusp‐like dependence of the heat capacity on the temperature, which for the infinite number of bosons transforms into the discontinuity of urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0011, is an indication of the phase transition to the condensate state. Some other physical characteristics such as the critical temperature urn:x-wiley:00033804:media:andp201600081:andp201600081-math-0012 and ground‐level population of the Bose‐Einstein condensate are calculated and analyzed as a function of the field and extrapolation length. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field.
Keywords:Electric field  quantum well  Robin boundary condition  Bose‐Einstein condensation  Fermi‐Dirac statistics  heat capacity
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