Hubbard equation

Hubbard and Lightfoot (HI la) earlier reported a Sc,/3 dependence on the basis of measurements in which the Schmidt number was varied over a very large range. The data did not exclude a lower Reynolds number exponent than 0.88, and reaffirmed the value of the classical Chilton-Colburn equation for practical purposes. Recent measurements on smooth transfer surfaces in turbulent channel flow by Dawson and Trass (D8) also firmly suggest a Sc13 dependence and no explicit dependence of k+ on the friction coefficient, with Sh thus depending on Re0,875. The extensive data of Landau... 

Equation 7.31 was derived from a least squares fit to the data given in W.N. Hubbard, D. W. Scott, G. Waddington. Standard States Corrections for Combustions inaBomb at Constant Volume. In Experimental Thermochemistry, vol. 1 F. D. Rossini, Ed. Interscience New York, 1956 p. 93. 

Hubbard and Miller87 used a Lewis acid catalyzed Diels-Alder reaction between y.y-disubstituted o. /i-unsaluralcd esters and cyclopentadiene in their approach toward oligomeric cyclopentanoids. In order for the reaction to proceed, they needed to add trimethylaluminum as a desiccant prior to addition of the Lewis acid catalyst aluminum trichloride. The endo/exo selectivity of the reaction with 97, depicted in equation 29, increased from 98/99 = 75/25 to 88/12 when the reaction temperature was dropped from room temperature to —20 °C. 

Equation (20) may be seen as the combination of the two processes of direct and inverse photoemission, when the 5 f shell retains a strong character of localization (in case of itinerant 5fs, the Hubbard model predicts that the kinetic energy due to itineracy creates statistical fluctuations between the polar states, so that the itinerant character is lost). 

For a transition of Mott type we shall show in Chapter 4, Section 3, neglecting the discontinuity resulting from long-range forces, that the transition should occur when 2zl = U. Near the transition the energy needed to excite an electron into the upper Hubbard band is U — 2zl. The wave function of an electron then falls off as e-fltr, where a=2m(I7 — 2zI)1/2/fc2. Thus the amount of spin in the sphere surrounding each atom will be made up from electrons on many of the surrounding atoms, and will clearly go to zero as [Pg.88]

Such spin polarons should not have a mass much greater than m in Si P. Moreover, they can pass freely from one atom to another, and are not impeded by the antiferromagnetic order. Thus the bandwidth of each Hubbard band should, we believe, still be of order 2zl, as it is for large values of U/B, and the equation... 

Fig. 5.13 Schematic illustration of how two Hubbard bands, with localized tails (shaded), resulting from disorder, can overlap, so that the equation (Bl+B2) U determines approximately the concentration at which the transition occurs, while the properties of the materials near the transition are those resulting from a transition of Anderson type.

Isotope ratios have been used with some success in the past to determine the importance of gas phase (Equation 4) verses aqueous phase (Equations 2A,2B,2C) oxidation of SO2. Saltzman et al. (24) compared the S34S values for SO2 and sulfate from samples collected from Hubbard Brook Experimental Forest (HBEF) in the non-urban northeastern US. They found discriminations which were intermediate to those expected for the individual oxidation mechanisms and suggested that both gas and aqueous phase oxidation were important. Newman (50) found that 6 S values for SO2 in the plume of an oil fired power plant decreased with distance (and time) from the stack which they attributed to equilibrium isotope effects. 

The aim of this work is to demonstrate that the above-mentioned unusual properties of cuprates can be interpreted in the framework of the t-J model of a Cu-O plane which is a common structure element of these crystals. The model was shown to describe correctly the low-energy part of the spectrum of the realistic extended Hubbard model [4], To take proper account of strong electron correlations inherent in moderately doped cuprate perovskites the description in terms of Hubbard operators and Mori s projection operator technique [5] are used. The self-energy equations for hole and spin Green s functions obtained in this approach are self-consistently solved for the ranges of hole concentrations 0 < x < 0.16 and temperatures 2 K< T <1200 K. Lattices with 20x20 sites and larger are used. 

Physics described by the model with so many parameters is very rich and the model is able particularly to treat heavy fermion systems. To study the model many approaches were suggested (see reviews [2-5]). They are successful for particular regions of the parameter space but no one is totally universal. In this paper we apply to PAM the generating functional approach (GFA) developed first by Kadanoff and Baym [6] for conventional systems and generalized for strongly correlated electron systems [7-10]. In particular it has been applied to the Hubbard model with arbitrary U in the X-operators formalism [10]. The approach makes it possible to derive equations for the electron Green s function (GF) in terms of variational derivatives with respect to fluctuating fields. 

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

It is remarkable that Eq.(17) has exactly the same form as in the Hubbard model, only instead of a bare hopping matrix element 112 now we have the induced hopping element A12. Because equations of motion for iF-operators for PAM and the Hubbard model have the same form (with the only change 112 — A12) the equation of motion for a d-electron GF in PAM has the same form as in the Hubbard model [10]. 

A GF (20) for PAM obeys the same equation of motion as for the Hubbard model [10], namely... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

Keywords Carbon nanotubes Hubbard model Continual approach Non-linear equations Regular solutions Soliton lattices... 

Fig. 5a. Such behavior is typical for soliton solutions of non-linear Schroedinger equation, which is closely connected to the model of Hubbard [16]. Also note the cycle in Fig. 4b, such portrait may further be the typical feature of soliton lattice appearance.

The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is comprised of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is ... 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

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