Electronic properties of metals at low temperatures

A many body theory of an electron gas is developed to find the internal and correlation energies at low but finite temperatures. The contribution from the first order exchange, second order (regular and anomalous) exchange, and ring diagrams are treated. The Fermi momentum and the correlation energy are determined as functions of the density by two different methods, one being based on iteration and the other a direct solution of the number density relation. It was found that the iterative solutions which are correct to ordere ² ore ⁴ become negative forr _s of order 5 while the direct solutions do not, indicating the invalidity of the former. Hence, the correlation energy evaluated to the same orders by iteration will not be satisfactory in the same range. The highest order iterative solution which includes terms of ordere ⁶ does not show such a breakdown. These terms which give the contribution of orderr _s to the correlation energy are therefore important and tend to reduce the magnitude of the correlation energy. The corresponding curve is indeed close to that determined by the direct method for smallr _s but a significant deviation takes place at largerr _s . The Coulomb interaction seems less effective at higher temperatures. The internal energy is also determined as a function of density and temperature.     

Correlation energy of 2-D electrons

The correlation energy and the Fermi momentum of an electron gas in 2-D are evaluated explicitly as functions of density. The ring diagram and first- and second-order exchange contributions are treated. In comparison with the 3-D case, the kinetic energy for the same r_s is approximately one-half and the exchange and correlation energies are somewhat larger. The ground state energy plotted against r_s shows a minimum at around r_s = 1.65 with a minimum value of ?0.9858 Ryd. If the third-order ring contribution is added, the curve is shifted upward. The correlation energy is ?0.6258 to order e⁴. The third-order ringw contribution increases this value almost linearly with r_s. The Fermi momentum decreases with r_s due to the contribution. Different from the 3-D case, no ln r_s term appears in the correlation energy within the approximation.     

On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀( n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^–2/3 , followed byr _s ^/–4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being –0.481915 Rydbergs.     

Rigorous evaluation of the ring diagram contribution to the ground state energy of an electron gas

The ground state energy and the correlation energy of an electron gas are evaluated rigorously without using the smallr _s expansion and the small momentum-transfer approximation in the ring diagram contribution and taking into consideration the first order and second order exchange graphs. The Fermi momentum is determined by solving the number density equation without using iteration and is compared with that obtained by iteration. The ground state energy is found to stay positive in contrast to the iterative solution which becomes negative beyond a certain value ofr _s .     

Thermodynamics of an interacting Fermi system in the static fluctuation approximation

We suggest a new method of calculation of the equilibrium correlation functions of an arbitrary order for the interacting Fermi-gas model in the framework of the static fluctuation approximation method. This method based only on a single and controllable approximation allows obtaining the so-called far-distance equations. These equations connecting the quantum states of a Fermi particle with variables of the local field operator contain all necessary information related to the calculation of the desired correlation functions and basic thermodynamic parameters of the many-body system. The basic expressions for the mean energy and heat capacity for the electron gas at low temperatures in the high-density limit were obtained. All expressions are given in the units of r _s , where r _s determines the ratio of a mean distance between electrons to the Bohr radius a ₀. In these expressions, we calculate terms of the respective order r _s and r _s ². It is also shown that the static fluctuation approximation allows finding the terms related to higher orders of the decomposition with respect to the parameter r _s .     

On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀(π n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^?2/3 , followed byr _s ^/?4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being ?0.481915 Rydbergs.     

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice. I

In 1969, Andreev and Lifshitz have conjectured the existence of a supersolid phase taking place at zero temperature between the quantum liquid and the solid. In this and a succeeding paper, we re-visit this issue for a few polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. This paper is restricted to the magic number of particles N = 4 for which a square Wigner molecule is formed when U increases and to the size L = 6 suitable for exact numerical diagonalizations. When the Coulomb energy to kinetic energy ratio r _s = UL/(2t ) reaches a value r _s ^F ≈ 10, there is a level crossing between ground states of different momenta. Above r _s ^F, the mesoscopic crystallization proceeds through an intermediate regime ( r _s ^F < r _s < r _s ^W ≈ 28) where unpaired fermions with a reduced Fermi energy co-exist with a strongly paired, nearly solid assembly. We suggest that this is the mesoscopic trace of the supersolid proposed by Andreev and Lifshitz. When a random substrate is included, the level crossing at r _s ^F is avoided and gives rise to a lower threshold r _s ^F(W) < r _s ^F where two usual approximations break down: the Wigner surmise for the distribution of the first energy excitation and the Hartree-Fock approximation for the ground state. Received 21 June 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: jpichard@cea.fr     

Ferromagnetism of the electron gas

The ground-state energy of the ferromagnetic electron gas is calculated for the relative polarizationζ=0−1 and the interelectron separationr _s =5−12. The method consists in describing the electron gas approximately by a quadratic boson Hamiltonian, and contains the random-phase approximation as a special case. Numerical studies show that in both the random-phase and the present approximations the paramagnetic state has the lowest energy: the energy increases withζ for all values ofr _s considered. In the present approximation instabilities are found to occur forr _s above a critical value, due to exchange processes of finite momentum transfers. Forζ=0 this critical value ofr _s is 9.4; it decreases with increasingζ. However, the fully-polarized state (ζ=1), which lies above the rest, is always stable. The conclusions are as follows: (1) Forr _s <9.4 the electron gas is paramagnetic. (2) Atr _s =9.4 it goes over to the fully-polarized ferromagnetic state. (3) This phase transition requires an energy absorption of 0.03 rydberg per electron. (4) The fully-polarized state is not obtainable as the limitζ→1.     

Higher order terms in the inflaton potential and the lower bound on the tensor to scalar ratio r

The MCMC analysis of the CMB + LSS data in the context of the Ginsburg–Landau approach to inflation indicated that the fourth degree double-well inflaton potential in new inflation gives an excellent fit of the present CMB and LSS data. This provided a lower bound for the ratio r of the tensor to scalar fluctuations and as most probable value r ? 0.05, within reach of the forthcoming CMB observations. In this paper we systematically analyze the effects of arbitrarily higher order terms in the inflaton potential on the CMB observables: spectral index n_s and ratio r. Furthermore, we compute in close form the inflaton potential dynamically generated when the inflaton field is a fermion condensate in the inflationary universe. This inflaton potential turns out to belong to the Ginsburg–Landau class too. The theoretical values in the (n_s, r) plane for all double well inflaton potentials in the Ginsburg–Landau approach (including the potential generated by fermions) fall inside a universal banana-shaped region B. The upper border of the banana-shaped region B is given by the fourth order double-well potential and provides an upper bound for the ratio r. The lower border of B is defined by the quadratic plus an infinite barrier inflaton potential and provides a lower bound for the ratio r. For example, the current best value of the spectral index n_s = 0.964, implies r is in the interval: 0.021 < r < 0.053. Interestingly enough, this range is within reach of forthcoming CMB observations.     

Comparative Studies of the Exchange-Correlation Potential Formulas for Coulomb Systems

The exchange-correlation part (xc) to the free energy is numerical evaluated in the RPA at arbitrary degree of degeneracy. The results are compared with numerical data of easy-to-use analytic fit-formulas or Padé approximants of the xc-term. All together results show very high accuracy at extremly high densities (r_s ≈ 1). The agreements disappear between the several formulas for increasing Brueckner parameter r_s. Numerical results for the xc-potentials (pressure and chemical potential) at finite temperatures for an electron-ion system are given. The xc-part of the ground state energy of our electron-ion model is compared with the ground state energy for metallic hydrogen and with Monte-Carlo calculations.     

Quantum chromodynamics corrections to polarised deep-inelastic electron-nucleon scattering

Quantum chromodynamics corrections to orderα _s (the running coupling constant), using the quark-parton approach are calculated for the spin-dependent structure functions in deep-inelastic polarised electron-nucleon scattering. Consequences of these corrections for the Bjorken sum rule and the asymmetry in the case of longitudinally polarised (with respect to the beam) nucleons is discussed which could provide possible tests of quantum chromodynamics. Comparison of our results with the moments of the flavour non-singlet contribution to the structure functions obtained using operator product expansion is also given. An erratum to this article is available at .     

Nuclear magnetic resonance spectra of deuterated methanes and 2,2-dimethylpropane in nematic liquid crystals

A simple variational approach is used to study the expectation values <r ₁₂ ^-1> and <r ₁₂ ²> for the two-electron 1s2p ¹ P and 1s2p ³ P. Several ‘anomalous’ phenomena are clearly displayed by the results, which provide some grounds for criticism of traditional arguments concerning spatial correlation between parallel spins.     

Some perturbation calculations for 1s2p, 1s 22p,and 1s 22s

The Dalgarno interchange theorem is used, together with the U function approach, to evaluate the first order perturbation corrections to <r ₁ ^-1+r ₂ ^-1> and <r ₁ + r ₂> for the two-electron states 1s2p ¹ P and 1s2p ³ P. The results for <r ₁ + r ₂> are extended by using a screening approximation, and are compared with the results of accurate variational calculations. The first order perturbation correction to spin-weighted expectation values of type <ΣV(r_j )s_zj > is given for the three-electron states 1s ²2p ² P and 1s ²2s ² S. The case V(r_j ) = r _j ^-1 is treated in detail.     

Bound state solutions of the two-electron Dirac-Coulomb equation

We present a variational method for solving the two-electron Dirac-Coulomb equation. When the expectation value of the Dirac-Coulomb Hamiltonian is made stationary for all possible variations of the different components of a well-behaved trial function one obtains solutions representative of the physical bound state wave functions. The ground state wave function is derived from the application of a minimax principle. Since the trial function remains well-behaved, the method remains safe from the twin demons of variational collapse and continuum dissolution. The ground state wave function thus derived can be interpreted as a linear combination of different configurations. In particular, the admixing of intermediate states having one (two) electron(s) deexcited to a negative-energy orbital (orbitals) contributes a second-order level shiftE ₀₋ ⁽²⁾ which can be identified with the second-order shift due to the Pauli blocking of the production of one (or two) virtual electron-positron pair(s). Thus the minimax solution corresponds to the renormalized ground state in quantum electrodynamics, with deexcitations to negative-energy orbitals taking the place of the avoidance of virtual pairs. If one extends the relativistic configuration interaction (RCI) treatment by additionally including negative-energy and mixed-energyeigenvectors of the Dirac-Hartree-Fock hamiltonian matrix in the two-electron basis, the calculated energy will be shifted from the conventional RCI value by an amount that is much smaller thanE ₀₋ ⁽²⁾ . For two-electron atoms, we have derived expressions for the all-spinor limit (δE) and thes-spinor limit (δE _s) of this shift in leading orders. The all-spinor limit (δE) is of orderα ⁴ Z ^{4 1/3} whereas thes-spinor limit (δE _s) is of orderα ⁴ Z ^{3 2/3}. leading components are related to the 1-pair component ofE ₀₋ ⁽²⁾ in a simple way, and the relationships offer the possibility of computing energy due to virtual pairs. Numerical results are discussed.     

Resonant electron impact excitation of highly charged Fe ions studied with a compact electron beam ion trap

We present extreme ultraviolet spectra of 3s3p–3s3d transitions in Fe¹⁴⁺ observed with a compact electron beam ion trap. The contributions of indirect excitation via a metastable state and resonant excitation are studied by observing the electron energy dependence of the spectra for the energy range of 60–210 eV. The results indicate that the 3s3d ¹D₂ level is directly excited from the 3s² ground state whereas the 3s3d ³D₃ level has a large contribution of the indirect excitation via the 3s3p ³P₂ metastable state. Comparisons with the theoretical excitation cross sections including MNn resonant excitations show good qualitative agreement with the experimental results for the electron energy dependent features.     

Effect of a lattice upon an interacting system of electrons in two dimensions: Breakdown of scaling and decay of persistent currents

The ground state of an electron gas is characterized by the interparticle spacing to the effective Bohr radius ratio r _s = a/a _B ^*. For polarized electrons on a two dimensional square lattice with Coulomb repulsion, we study the threshold value r _s ^* below which the lattice spacing s becomes a relevant scale and r _s ceases to be the scaling parameter. For systems of small ratios s/a _B ^*, s becomes only relevant at small r _s (large densities) where one has a quantum fluid with a deformed Fermi surface. For systems of large s/a _B ^*, s plays also a role at large r _s (small densities) where one has a Wigner solid, the lattice limiting its harmonic vibrations. The thermodynamic limit of physical systems of different a _B ^* is qualitatively discussed, before quantitatively studying the lattice effects occurring at large r _s . Using a few particle system, we compare exact numerical results obtained with a lattice and analytical perturbative expansions obtained in the continuum limit. Three criteria giving similar values for the lattice threshold r _s ^* are proposed. The first one is a delocalization criterion in the Fock basis of lattice site orbitals. The second one uses the persistent current which can depend on the interaction in a lattice, while it becomes independent of the interaction in the continuum limit. The third one takes into account the limit imposed by the lattice to the harmonic vibrations of the electron solid.Received: 20 January 2004, Published online: 18 June 2004PACS: 71.10.-w Theories and models of many-electron systems - 71.10.Fd Lattice fermion models (Hubbard model, etc.) - 73.20.Qt Electron solids     

Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b ? a _B are studied in random phase approximation (RPA) and for the less investigated region r _s < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by ? _c (b, r _s ) = α(r _s )/b + β(r _s ) ln(b) + η(r _s ), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.     

Resonant acceptor states and terahertz stimulated emission of uniaxially strained germanium

The stimulated emission spectrum of uniaxially strained p-Ge is presented. The energy spectrum of the states of a shallow acceptor in Ge under uniaxial compression is calculated. The threshold pressure at which the acceptor state split off from the ground state becomes resonant is found. The pressure dependence of the width of this resonant level is calculated. The stimulated emission lines are identified. In particular, it is shown that the principal emission peak corresponds to the transition of holes from the resonant 1s (1s _r) state to the local p _±1 state. The probabilities of optical transitions are calculated. A mechanism of population inversion due to the intense resonant scattering of hot holes with an energy corresponding to the position of the 1s _r level is proposed. Zh. éksp. Teor. Fiz. 115, 89–100 (January 1999)     

Effects of the reduction of the dimension of a system upon spin ordering in a degenerate electron gas

Collective effects of spin ordering in a quasi-one-dimensional degenerate electron gas are discussed. The total energy of the quasi-one-dimensional system, as well as the exchange contribution per particle, has been calculated by the Hartree-Fock method. It has been shown that spontaneous polarization can be observed in the system when a universal parameter related to the density of the particles satisfies the inequality r _s ≥ 0.476. A comparative analysis of one-, two-, and three-dimensional systems has been performed. A general expression has been obtained for the total energy per particle as a function of the degree of polarization and dimension of the system. According to this expression, the possibility of spontaneous polarization in the system is closely correlated with the dimension of the system.