Two-pole Approximation Research Articles

Time-dependent self-diffusion coefficient of interacting Brownian particles.

We study the time-dependent self-diffusion coefficient for a suspension of interacting, spherical Brownian particles. Guided by an exact result for a dilute suspension of hard spheres, we conjecture that the Fourier transform of the memory function may be represented as a meromorphic function of the square root of the frequency. We show that a two-pole approximation provides a suitable framework for the analysis of experimental data.

Tracer diffusion in cubic lattices

The multivariable Zwanzig-Mori formalism is used to study the tracer diffusion by nearest-neighbor jumps in cubic regular lattices. The effect of the nearest-neighbor correlations has been calculated exactly while the long-range correlations are computed within mode-coupling theory. The agreement between our results and the Monte Carlo results for the tracer correlation factor lies within 3% in the whole concentration range. An alternative method, the continued-fraction expansion of Mori, is also used. It has been shown that for self-diffusion the two-pole approximation is equivalent to the multivariable theory. Furthermore, through the successive steps in the continued-fraction, correlation effects from the first, second, and third steps of the jump of the tracer particle are also estimated.

Correlation theory applied to the static and dynamic properties of EuO and EuS

The paramagnetic scattering was recently measured for EuO.1) It was found that spin-wave-like excitations develop for wave vectors approaching the zone boundary. The spectrum was found to be well described by damped harmonic oscillators (also called the two-pole approximation). This approximation was used previously in the correlation theory2) primarily to calculate static properties. Self-consistent dynamic and static calculations have here been performed for EuO, which is an ideal Heisenberg magnet with significant second-nearest-neighbor interaction (J2 = J1/5). The two-pole approximation describes accurately the correlation range, the static susceptibility, and the qualitative behavior of the dynamic properties (i.e., the wave vector at which peaks appear in the spectrum as a function of temperature). However, in order to also obtain the correct frequency scale it is necessary to use a cutoff of the spectrum at high frequencies, which cannot be seen experimentally, but which significantly influences the frequency moments. It was found that the finite J2 has significant importance for a comparison between theory and experiment. It is concluded that the calculation for a simple cubic n.n. magnet by Hubbard3) does not describe the EuO data accurately, neither with respect to line shape nor frequency scale. Significant differences are to be expected between EuO and EuS having opposite sign for J2.

New two-pole approximation for the plasma dispersion function Z

A new two-pole approximation for the plasma dispersion function Z (s) is obtained by a modified Padé (1/2) approximant. It gives a better agreement than the known two-pole approximation. It also has the advantage that the same poles are used for Z (s) and its derivative Z′ (s).

Electron Correlation in Narrow Energy Bands. II. One Reversed Spin in an Otherwise Fully Aligned NarrowSBand

In a previous paper, a new Green's-function decoupling scheme was applied to the Hubbard Hamiltonian, and an improved version of Hubbard's first approximation was obtained. That result did not reduce to the correct low-density limit as obtained by Kanamori. In the present article, the theory is improved for the special case of a single reversed spin in an otherwise fully aligned band, and the improved theory is correct in the low-density limit. Numerical results are presented for the simple cubic lattice. If we define an effective exchange-interaction parameter ${U}_{\mathrm{eff}}$ as the $k=0$ reversed-spin self-energy for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, divided by the number $n\ensuremath{\uparrow}$ of up-spin electrons per site, we find that the present result departs rather rapidly from the Kanamori result as $n\ensuremath{\uparrow}$ is increased, and it is concluded that the Kanamori result overestimates the increase in ${U}_{\mathrm{eff}}$ with $n\ensuremath{\uparrow}$, at least in the present case. For intermediate values of $n\ensuremath{\uparrow}$, the two-pole approximation of the previous article and the present calculation give very similar results for this quantity.

Electron Correlation in Narrow Energy Bands. I. The Two-Pole Approximation in a NarrowSBand

An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's $T$-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.

Two-Pole Approximation for the Plasma Dispersion Function

A simple algebraic approximation to the Hilbert transform of the Gaussian, valid over most of the complex plane, is given.

Determination of Pion-NucleonS-Wave Scattering Lengths by theNDMethod

The pion-nucleon $S$-wave scattering lengths are calculated using the method originally developed by Bal\'azs, wherein an effective-range two-pole approximation is made for the numerator function. The residues of the effective-range poles are determined by matching the amplitude and its derivative with those calculated with a fixed energy dispersion relation. In calculating the latter the contribution of only the ${N}^{*}$ and $\ensuremath{\rho}$, together with those of appropriate nucleon-pole terms, are retained. The calculated scattering length in the $T=\frac{3}{2}$ state is in excellent agreement with experimental result, while for the $T=\frac{1}{2}$ state, the calculated value, though of the right sign, is about twice the experimental scattering length.