First and Second Derivatives of the Chemical Potential for Noninteracting Particles
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First and Second Derivatives of the Chemical Potential for Noninteracting Particles Jacob Katriel1,2 · Hongrui Zhang1 Received: 6 July 2020 / Accepted: 3 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Both the first and the second derivatives of the chemical potential with respect to the temperature, for any system of noninteracting bosons, are shown to be negative. For noninteracting fermions, these derivatives can be either negative or positive, but for generic spectra they are both negative at higher temperatures. Keywords Noninteracting bosons · Chemical potential
1 Introduction In the grand canonical treatment of a system of non-interacting indistinguishable particles, the number of particles and the chemical potential are related by [1, 2]
N=
∑ i
exp
(
1 ei −𝜇 kT
)
,
(1)
±1
where the plus sign refers to fermions and the minus sign to bosons. In the case of bosons, if the energy spectrum is such that a transition to a Bose–Einstein condensate takes place at some temperature Tc [3], then the discussion that follows applies only above this temperature. The class of systems of noninteracting particles is much broader and richer than the ideal gases, which involve very definite spectra that depend on the dimension of the space considered. Several approaches to the calculation of the chemical potential, 𝜇 , given the temperature, T, the number of particles, N and the one-particle energies, e1 ≤ e2 ≤ e3 ≤ ⋯ , have very recently been reviewed by Cowan [4], who presented explicit expressions and detailed plots for the
* Jacob Katriel [email protected] 1
Guangdong Technion - Israel Institute of Technology, Shantou, Guangdong, China
2
Department of Chemistry, Technion - Israel Institute of Technology, Haifa 32000, Israel
13
Vol.:(0123456789)
Journal of Low Temperature Physics
chemical potentials of ideal Fermi, Bose and Maxwell gases in one, two and three dimensions. In Sect. 2, it is demonstrated that for a system of noninteracting fermions the first and the second derivatives of the chemical potential with respect to the temperature can be either positive or negative at low temperature. In the former case, both become negative (first the second derivative, then the first) as the temperature rises. In Sects. 3 and 4, it is shown that for any system of noninteracting bosons both the first and the second derivative of the chemical potential with respect to the temperature are always negative.
2 The First and Second Derivatives of the Chemical Potential for Noninteracting Fermions For an ideal Fermi gas, the Sommerfeld expansion yields a vanishing first derivative of the chemical potential, at T → 0 . At the same limit, the second derivative is positive for the one-dimensional and negative for the three-dimensional free electron gas, breaking down in two dimensions [4]. More generally, the Sommerfeld expansion yields [5]
𝜇 = 𝜖F −
g� (𝜖F ) 𝜋2 , (kB T)2 6 g(𝜖F )
where g(𝜖) is the energy-density of states. Hence, the sign of the second deri
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