Computation of a general integral of Fermi–Dirac distribution by McDougall–Stoner method

Fukushima, Toshio

doi:10.1016/j.amc.2014.04.028

Cited by 12 publications

(6 citation statements)

References 29 publications

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“…It is worth remarking that eq 11 cannot be solved analytically at all temperatures, due to the presence of Fermi integrals f p , in which the parameter a is unknown. In regard to the interest of the Fermi integrals, their analytic approximations attracted several researchers. − In this work, rather than using proposed approximation methods to evaluate Fermi integrals and derive the entropy, we used a numerical approach similar to the one proposed by Bartmess. , This approach is an iterative and self-consistent procedure based on the computed temperature as defined in eq . Analytic results for special cases (completely degenerate and nondegenerate gas) have been discussed below.…”

Section: Formalism Of a Free Quantum Gasmentioning

confidence: 99%

“…The completely degenerate gas is observed for a → ∞, which corresponds to T → 0. In this case, the famous Sommerfeld expansion for the Fermi integrals is known to be very accurate. − On the basis of Sommerfeld’s ideas, we provided the following explicit expressions of the TPP at very low temperatures. Planting normalΛ ( T ) = true( 6 π 2 g true) 2 / 3 ℏ 2 m ρ false( T false) 2 / 3 with ρ( T ) = N / V the proton density of the considered gas at a temperature T , the Fermi energy can be defined as ε F = Λ(0).…”

Section: Formalism Of a Free Quantum Gasmentioning

confidence: 99%

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Revision of the Thermodynamics of the Proton in Gas Phase

2014

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Proton transfer is ubiquitous in various physical/chemical processes, and the accurate determination of the thermodynamic parameters of the proton in the gas phase is useful for understanding and describing such reactions. However, the thermodynamic parameters of such a proton are usually determined by assuming the proton as a classical particle whatever the temperature. The reason for such an assumption is that the entropy of the quantum proton is not always soluble analytically at all temperatures. Thereby, we addressed this matter using a robust and reliable self-consistent iterative procedure based on the Fermi-Dirac formalism. As a result, the free proton gas can be assumed to be classical for temperatures higher than 200 K. However, it is worth mentioning that quantum effects on the gas phase proton motion are really significant at low temperatures (T ≤ 120 K). Although the proton behaves as a classical particle at high temperatures, we strongly recommend the use of quantum results at all temperatures, for the integrated heat capacity and the Gibbs free energy change. Therefore, on the basis of the thermochemical convention that ignores the proton spin, we recommend the following revised values for the integrated heat capacity and the Gibbs free energy change of the proton in gas phase and, at the standard pressure (1 bar): ΔH0→T = 6.1398 kJ mol(-1) and ΔG0→T = -26.3424 kJ mol(-1). Finally, it is important noting that the little change of the pressure from 1 bar to 1 atm affects notably the entropy and the Gibbs free energy change of the proton.

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Section: Formalism Of a Free Quantum Gasmentioning

confidence: 99%

Section: Formalism Of a Free Quantum Gasmentioning

confidence: 99%

Revision of the Thermodynamics of the Proton in Gas Phase

2014

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“…In order to ensure and accelerate their process of construction, a set of the quadruple precision approximations of the integrals are first developed by combining (i) the optimally truncated Sommerfeld expansion [30], and (ii) the piecewise truncated Chebyshev series expansion [36], as well as (iii) the reflection formula for F k ðgÞ of integer orders and positive arguments [17]. The relative errors of the new minimax approximations amount to 3-13 machine epsilons at most and to 0-4 machine epsilons in average.…”

Section: Resultsmentioning

confidence: 99%

“…1. This is true even for an extension of the numerical integration method of McDougall and Stone [30,Fig. 14].…”

Section: Existing Methods To Compute Fermi-dirac Integralsmentioning

confidence: 99%

“…During the preparation of an extension of the numerical quadrature method developed for F 1=2 ðgÞ by McDougall and Stoner [2, Section 3] to a general integral of the Fermi-Dirac distribution [30], it was noticed that the computational speed of the approximations developed by Macleod [28] is not so fast as expected, namely not as extrapolated from the results of the rational approximations of lower accuracies [18,24,27]. Compare the averaged CPU times of the methods of Trellakis et al and Macleod[27,28] listed in Table 4.…”

Section: Background Of Present Articlementioning

confidence: 99%

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