Interpolation formulas for Fermi-Dirac functions

Калиткин, Н. Н.; Kuzmina, L. V.

doi:10.1016/0041-5553(75)90088-9

Cited by 7 publications

(8 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our computational scheme is similar to that of [3,10]. We also use the GNU scientific library [11] to calculate the Fermi-Dirac integrals and integrate ordinary differential equations with adaptive step-size control.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…(37) Thus, we want to stress that the approach we propose helps one to determine all necessary thermodynamic properties with a given accuracy. This can be done by calculating the integrals in ( 4), ( 5), ( 23), ( 24), ( 31) and (33) as a solution of a system of differential equations together with the TF equation ( 2) [10]. To do this, we replace the integral b a f (x) dx by the Cauchy problem dg/dx = f (x), g(a) = 0, with the value of the integral being g(b).…”

Section: Thermodynamic Functions At θ →mentioning

confidence: 99%

See 1 more Smart Citation

Thermal contribution to thermodynamic functions in the Thomas–Fermi model

Shemyakin¹,

Levashov²,

Obruchkova³

et al. 2010

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We propose a method of calculation of thermodynamic functions in the Thomas–Fermi model at finite temperature θ. Expressions for first and second derivatives of the free energy are analytically obtained in the framework of the model. A special treatment of thermodynamic functions at low temperatures is provided by asymptotic series expansion at θ → 0. A special algorithm is used to ensure required accuracy for all values in a wide range of volumes and temperatures. We compare the results of our computations with ideal Boltzmann and Fermi gas models.

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

Section: Thermodynamic Functions At θ →mentioning

confidence: 99%

Thermal contribution to thermodynamic functions in the Thomas–Fermi model

Shemyakin¹,

Levashov²,

Obruchkova³

et al. 2010

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…The EOS provides for a correct description of melting and evaporation, as well as the ionization effects. The Thomas-Fermi model with quantum and exchange corrections [14,15] is used as the asymptotic at high values of the compression ratio ρ/ρ s .…”

Section: Problem Statement and Numerical Methodsmentioning

confidence: 99%

Symmetrically converging plane thermonuclear burn waves

Charakhch’yan¹,

Khishchenko²

2013

Plasma Phys. Control. Fusion

View full text Add to dashboard Cite

Five variants of a one-dimensional problem on synchronous bilateral action of two identical drivers on opposite surfaces of a plane layer of DT fuel with the normal or five times greater initial density, where the solution includes two thermonuclear burn waves propagating to meet one another at the symmetry plane, are simulated. A laser pulse with total absorption of energy at the critical density (in two variants) and a proton bunch that provides for a nearly isochoric heating (in three variants) are considered as drivers. A wide-range equation of state for the fuel, electron and ion heat conduction, self-radiation of plasma and plasma heating by α-particles are taken into account.In spite of different ways of ignition, various models of α-particle heat, whether the burn wave remains slow or transforms into the detonation wave, and regardless of way of such a transformation, the final value of the burn-up factor depends essentially on the only parameter Hρ 0 , where H is the half-thickness of the layer and ρ 0 is the initial fuel density. This factor is about 0.35 at Hρ 0 ≈ 1 g cm −2 and about 0.7 at Hρ 0 ≈ 5 g cm −2 . The expansion stage of the flow (after reflecting the burn or detonation wave from the symmetry plane) gives the main contribution in forming the final values of the burn-up factor and the gain at Hρ 0 ≈ 1 g cm −2 and increases them approximately two times at Hρ 0 ≈ 5 g cm −2 . In the case of the proton driver, the final value of the gain is about 200 at Hρ 0 ≈ 1 g cm −2 and about 2000 at Hρ 0 ≈ 5 g cm −2 . In the case of the laser driver, the above values are four times less in conformity with the difference between the driver energies.

show abstract

“…is needed, where the coefficients b 2 and b 1 are determined from consideration of the Thomas-Fermi model with quantum and exchange corrections [42][43][44][45][46]:…”

Section: Eos Modelmentioning

confidence: 99%

“…In this paper, a model of the EOS for matter in the form of a function of pressure P = P (V, E) is developed. In contrast to the previously known EOSs (for bismuth) [32][33][34][35][36][37][38][39][40][41], a new expression is proposed for the internal energy of matter on the zero-temperature isotherm (cold curve) in a wide range of densities (ρ = V −1 ), which has asymptotics to the Thomas-Fermi model with quantum and exchange corrections [42][43][44][45]. A new EOS is developed for a body-centered cubic (bcc) solid phase and a melt of bismuth in the highpressure region.…”

Section: Introductionmentioning

confidence: 99%

Equation of State for Bismuth at High Energy Densities

Khishchenko

2022

Energies

View full text Add to dashboard Cite

The purpose of this work is to describe the thermodynamic properties of bismuth in a broad scope of mechanical and thermal effects. A model of the equation of state in a closed form of the functional relationship between pressure, specific volume, and specific internal energy is developed. A new expression is proposed for the internal energy of a zero-temperature isotherm in a wide range of compression ratios, which has asymptotics to the Thomas–Fermi model with corrections. Based on the new model, an equation of state for bismuth in the region of body-centered cubic solid and liquid phases is constructed. The results of calculating the thermodynamic characteristics of these condensed phases with the new EOS are compared with the available experimental data for this metal in waves of shock compression and isentropic expansion. The parameters of shock waves in air obtained earlier by unloading shock-compressed bismuth samples are reconsidered. The newly developed equation of state can be used in modeling various processes in this material at high energy densities.

show abstract

Interpolation formulas for Fermi-Dirac functions

Cited by 7 publications

References 5 publications

Thermal contribution to thermodynamic functions in the Thomas–Fermi model

Thermal contribution to thermodynamic functions in the Thomas–Fermi model

Symmetrically converging plane thermonuclear burn waves

Equation of State for Bismuth at High Energy Densities

Contact Info

Product

Resources

About