The computation of Fermi-Dirac functions

McDougall, Jane; Stoner, Edmund C.

doi:10.1098/rsta.1938.0004

Cited by 420 publications

(53 citation statements)

References 2 publications

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“…Throughout the paper we use the superscripts ± to refer to quantities pertaining to the half wells W ± , respectively, and also use ∆ to refer to the difference between the two quantities such as (13). However, we will often omit these superscripts for brevity as long as the distinction is unnecessary in the argument.…”

Section: Quantum Force On a Partition And Its Temperature Behavioursmentioning

confidence: 99%

“…We are thus forced to apply some approximations, but the standard approach of approximating the sum with an integral turns out to be insufficient to get the accuracy needed in the present situation, leaving us to work out novel methods for the present problem. For instance, the FermiDirac integral, which is often used to approximate the sum and is related to the Lerch transcendent, can be expressed via an asymptotic series [13], but the truncations of the series do not provide enough precision to recover the force difference obtained numerically. As we shall see soon, our approximations developed in this paper and performed partly with the help of numerical solutions for transcendental equations provide formulae which fit the numerical results reasonably well.…”

Section: Quantum Force On a Partition And Its Temperature Behavioursmentioning

confidence: 99%

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Boundary effect of a partition in a quantum well

Fülöp¹,

Tsutsui²

2007

J. Phys. A: Math. Theor.

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Abstract.The paper wishes to demonstrate that, in quantum systems with boundaries, different boundary conditions can lead to remarkably different physical behaviour. Our seemingly innocent setting is a one dimensional potential well that is divided into two halves by a thin separating wall. The two half wells are populated by the same type and number of particles and are kept at the same temperature. The only difference is in the boundary condition imposed at the two sides of the separating wall, which is the Dirichlet condition from the left and the Neumann condition from the right. The resulting different energy spectra cause a difference in the quantum statistically emerging pressure on the two sides. The net force acting on the separating wall proves to be nonzero at any temperature and, after a weak decrease in the low temperature domain, to increase and diverge with a square-root-of-temperature asymptotics for high temperatures. These observations hold for both bosonic and fermionic type particles, but with quantitative differences. We work out several analytic approximations to explain these differences and the various aspects of the found unexpectedly complex picture.

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Section: Quantum Force On a Partition And Its Temperature Behavioursmentioning

confidence: 99%

Section: Quantum Force On a Partition And Its Temperature Behavioursmentioning

confidence: 99%

Boundary effect of a partition in a quantum well

Fülöp¹,

Tsutsui²

2007

J. Phys. A: Math. Theor.

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“…For extreme degeneracy, D~I, Yj approaches D and I'lf approaches (3/2D). In this case the polarization of the electron gas is negligible compared with that of the nuclear gas: For intermediate values of D, Yj and then I'lf were obtained from (22) and (23), using the numerical tabulation of the Fermi-Dirac function given by McDougall and Stoner (1938). The results of these numerical calculations are given in Figure 1, the quantity I'(Yj)lf(Yj) being plotted against D.…”

Section: (23)mentioning

confidence: 99%

Electron Screening and Thermonuclear Reactions

Salpeter¹

1954

Aust. J. Phys.

601

569

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SummaryIn the interior of stars most atoms are ionized, but the electrostatic potential of a bare nucleus induces a spherically symmetric polarization of the surrounding electrons and nuclei. The effect of this screening charge cloud on the rate of thermonuclear reactions is investigated for the case of complete ionization of all atoms.The charge distribution and potential of the screening cloud is calculated for two limiting cases where the electrostatic interaction energy between neighbouring nuclei is small or large compared with the thermal energy (weak or strong screening). The charge cloud is also investigated for intermediate strength of screening, for nuclear species which are rare and have a large charge.Under most stellar conditions the impact parameter for a thermonuclear collision is much smaller than the radius of a screening cloud. For such cases, a simple formula is given relating the increaae in the reaction rate to the potential of the screening cloud. Numerical values are presented for a few typical reactions. For conditions typical for the interior of ordinary main sequence stars the increase in the reaction rate is fairly small, usually less than a factor of two.

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“…The F n (x) are the standard Fermi integrals; their properties are discussed on pp. 332-334 of [10] and their values for various n are tabulated in [11,12]. [7].…”

Section: Constructing the Eosmentioning

confidence: 99%

Test of a theoretical equation of state for elemental solids and liquids

2003

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We propose a means for constructing highly accurate equations of state (EOS) for elemental solids and liquids essentially from first principles, based upon a particular decomposition of the underlying condensed matter Hamiltonian for the nuclei and electrons. We also point out that at low pressures the neglect of anharmonic and electronphonon terms, both contained in this formalism, results in errors of less than 5% in the thermal parts of the thermodynamic functions. Then we explicitly display the forms of the remaining terms in the EOS, commenting on the use of experiment and electronic structure theory to evaluate them. We also construct an EOS for Aluminum and compare the resulting Hugoniot with data up to 5 Mbar, both to illustrate our method and to see whether the approximation of neglecting anharmonicity et al. remains viable to such high pressures. We find a level of agreement with experiment that is consistent with the low-pressure results.

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The computation of Fermi-Dirac functions

Cited by 420 publications

References 2 publications

Boundary effect of a partition in a quantum well

Boundary effect of a partition in a quantum well

Electron Screening and Thermonuclear Reactions

Test of a theoretical equation of state for elemental solids and liquids

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