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CRC Press eBooks

The extended Thomas-Fermi model

M. Brack, Rajat K. Bhaduri

Published Mar 5, 2018• 0 citations
Publisher

Abstract

In the application of the extended Thomas-Fermi (ETF) method to a many-fermion system, only those terms in the ft expansion will be included that give finite contribution to the energy. The ETF model yields the smooth part of the level density and other quantities. It will be derived from the E. Wigner-J. G. Kirkwood. It discusses the ETF density variational method and its use for obtaining in a self-consistent way the average parts of the energy and other observables of a many-fermion system in its ground state. Finally, a numerical smoothing of the level density may also be done, by replacing each delta function spike by an appropriate smoothing function, like a Gaussian or a Lorentzian. Since the TF approximation is obtained by replacing the sum over the quantum states by the integral over classical phase space, it is not surprising that it amounts to taking the inverse Laplace transform of the classical canonical partition function.

DOI
10.1201/9780429502828-4

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In the application of the extended Thomas-Fermi (ETF) method to a many-fermion system, only those terms in the ft expansion will be included that give finite contribution to the energy. The ETF model yields the smooth part of the level density and other quantities. It will be derived from the E. Wigner-J. G. Kirkwood. It discusses the ETF density variational method and its use for obtaining in a self-consistent way the average parts of the energy and other observables of a many-fermion system in its ground state. Finally, a numerical smoothing of the level density may also be done, by replacing each delta function spike by an appropriate smoothing function, like a Gaussian or a Lorentzian. Since the TF approximation is obtained by replacing the sum over the quantum states by the integral over classical phase space, it is not surprising that it amounts to taking the inverse Laplace transform of the classical canonical partition function.
Semiclassical theories such as the Thomas-Fermi and Wigner-Kirkwood methods give a good description of the smooth average part of the total energy of a Fermi gas in some external potential when the chemical potential is varied. However, in systems with a fixed number of particles NNN, these methods overbind the actual average of the quantum energy as NNN is varied. We describe a theory that accounts for this effect. Numerical illustrations are discussed for fermions trapped in a harmonic oscillator potential and in a hard-wall cavity, and for self-consistent calculations of atomic nuclei. In the latter case, the influence of deformations on the average behavior of the energy is also considered.
In this thesis, following L. Salasnich and F. Toigo [Phys. Rev. A 78, 053626 (2008)] we use an extended Thomas-Fermi (TF) density functional to describe a gas of interacting fermions composed of an even number N of particle in the unitary regime confined by a spherically symmetric harmonic trap. The great advantage in taking such a functional, relative to functionals based on single-fermion wave functions, is to depend only on the density number n and on its derivatives, so that the numerical load is independent of the number of particles. By minimizing the ETF functional with respect to the density we obtain a time-independent non-linear Schrodinger equation which we solve using the imaginary time propagation method, by integrating numerically the corresponding time-dependent Schrodinger equation. We determine the two characteristic parameters of the functionals ξ and λ by fitting the Monte Carlo total energy data obtained by D. Blume et al. [Phys. Rev. Lett. 99, 233201 (2007)]. We obtain the values ξ = 0.469 and λ = 0.082 leading to a χ2 = 42. These values are to be compared with ξ = 0.455 and λ = 0.13, leading to a χ2 = 282 as proposed by Salasnich and Toigo. We also study the dependence of the ground-state density profile on the values of the two parameters, finding that ξ enhances the width of the profile, while increasing λ smooths out the outer boundary of the fermionic cloud. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where nnn is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number NNN of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.
Using the $\ensuremath{\hbar}$ expansion of the Green's function of the Hartree-Fock-Bogoliubov equation, we extend the second-order Thomas-Fermi approximation to generalized superfluid Fermi systems by including the density-dependent effective mass and the spin-orbit potential. We first implement and examine the full correction terms over different energy intervals of the quasiparticle spectra in calculations of finite nuclei. Final applications of this generalized Thomas-Fermi method are intended for various inhomogeneous superfluid Fermi systems.