Nonoverlapping Potentials Research Articles

Bardeen's tunneling theory applied to intraorbital and interorbital hopping integrals between dopants in silicon

We utilize Bardeen's tunneling theory to calculate intra- and interorbital hopping integrals between phosphorus donors in silicon using known orbital wave functions. While the two-donor problem can be solved directly, the knowledge of hoppings for various pairs of orbitals is essential for constructing multi-orbital Hubbard models for chains and arrays of donors. To assure applicability to long-range potentials, we rederive Bardeen's formula for the matrix element without assuming non-overlapping potentials. Moreover, we find a correction to the original expression allowing us to use it at short distances. We also show that accurate calculation of the lowest donor-pair eigenstates is possible based on these tunnel couplings, and we characterize the obtained states. The results are in satisfactory quantitative agreement with those obtained with the standard H\"uckel tight-binding method. The calculation relies solely on the wave functions in the barrier region and does not explicitly involve donor or lattice potentials, which has practical advantages. We find that neglecting the central correction potential in the standard method may lead to qualitatively incorrect results, while its explicit inclusion raises severe numerical problems, as it is contained in a tiny volume. In contrast, using wave functions obtained with this correction in the proposed method does not raise such issues. Nominally, the computational cost of the method is to calculate a double integral along the plane that separates donors. For donor separation in directions where valley interference leads to oscillatory behavior, additional averaging over the position of the integration plane is needed. Despite this, the presented approach offers a competitive computational cost as compared to the standard one. This work may be regarded as a benchmark of a promising method for calculating hopping integrals in lattice models.

Landauer resistance in the case of multichannel scattering

The Landauer resistance ρ L has been generalized to the case of multichannel scattering of a particle by the system of N nonoverlapping random potentials that are localized at the points x i (i = 1, 2,..., N) and depend on x − x i and y. It has been shown that, in this case, a new resistance ρ L appears, which is an exponential function of N. The recurrence equation for determining the Landauer resistance ρ L has been derived and its solution in the general case has been obtained.

Electronic-structure method for general space-filling cell potentials.

It is shown that the scattering matrix in the form describing multiple-scattering processes for collections of muffin-tin (MT) potentials is rigorously valid for nonoverlapping potentials of arbitrary shape. In particular, the near-field corrections (NFC's) ostensibly arising in the latter case are shown to vanish identically. It follows that the well-known method of Korringa and Kohn and Rostoker for calculating electronic structure is valid for non-MT potentials. Model calculations illustrating directly the vanishing of NFC's and a calculation for elemental Nb are presented and the effects of convergence in such calculations are discussed.

Multiple-scattering theory beyond the muffin-tin approximation

The author presents a new derivation of multiple-scattering theory as applied in solid state physics. The present approach is a generalisation of the well known Korringa, Kohn and Rostoker (KKR) formalism and holds also for overlapping potentials. The resulting equations have the same structure as the KKR equations, except for extra energy integrations. It is shown that for non-overlapping potentials these integrations can be performed analytically. Then, they reduce to the KKR equations, in which all matrices are evaluated at one and the same energy. The introduction of a muffin-tin potential appears not to be essential in this reduction. From this, it is concluded that the so-called 'near-field' corrections, introduced in previous papers dealing with non-muffin-tin potentials, can be ignored.

Multiple-scattering theory for clusters of nonoverlapping potentials of arbitrary shape.

It is shown that the full scattering matrix associated with an assembly of nonoverlapping potentials of arbitrary convex shape has precisely the same form as the corresponding expression for potentials confined inside nonoverlapping spheres. A specific implication of this result is that the electronic structure of periodic solids can be obtained from the solutions of a secular equation formally identical to that of the Korringa-Kohn-Rostoker method for periodic muffin-tin potentials.

Non-eikonal corrections in potential scattering on two centres

Scattering of a spinless projectile from a system of two (over-lapping or non-overlapping) local potentials is studied in the first-order eikonal expansion. Corrections to the Glauber formula are found to come mainly from the non-eikonal propagation in the “direct” double scattering, and not from multiple scattering (“reflections”).A comparison is made between the exact (to the leading order) and two approximate results: (i) based on the theory of scattering from non-overlapping potentials, and (ii) neglecting the reflection terms and adopting the local “Born-like” off-shell extrapolation of the two-body amplitudes. The second approximation is found to be more accurate and technically simpler in calculation of multiple scattering from many-body systems.

The Lloyd formula and the muffin-tin CPA local density of states

The Lloyd formula (1967) for the density of states (DOS) of a finite cluster on nonoverlapping potentials in free space is derived in a direct way by local integration of the Green function of the system in the r representation. Similarly a new expression is derived from the DOS per atom in a disordered binary substitutional alloy, which is valid within the muffin-tin single-site coherent potential approximation. It is shown that this expression is equivalent to the one that is usually derived from the Lloyd formula, but that its numerical evaluation is simpler. Alternative expressions for the component DOS are also discussed.

The pion-deuteron forward elastic amplitude in the non-overlapping potentials model

The pion deuteron forward elastic amplitude has been calculated in the nonoverlapping potentials model which enables a description of off-shell propagation effects in terms of on-shell amplitudes. Calculations include spin, isospin and deuteron D-state probability effects. Two energy regions are considered. First the pion-nucleon P33 resonance region, where, using a formalism developed by Agassi and Gal, the full multiple-scattering series is summed in an approximation of P33 wave dominance of the higher-order scatterings. Second, for the subsequent higher-energy region, the double-scattering term only is calculated. Fermi smearing effects are included in both cases. Predictions for the total cross section, its dependence on the deuteron alignment and the real part of the forward elastic amplitude are compared with those of Glauber theory, and data where available. Convergence of the multiple-scattering series is also discussed.

Cluster Decomposition of the S Matrix and Density Expansion of Nuclear Phase Shifts

A systematic way of improving the Glauber multiple-scattering theory is discussed from the standpoint of the cluster decomposition of the S matrix. Since the decomposition is worked out without appealing to any specific approximations, the result is formally exact. Prescriptions which specify the order of each cluster contribution are considered (Z-ordering) in order to minimize the error arising when one introduces truncations in the theoretical scheme. Due to this Z-ordering technique, the convergence of the scheme seems very rapid especially for non-overlapping potentials. The structure and some basic properties of the two-body clusters are discussed in some detail. From the cluster-factorized expression for the S matrix, nuclear-density expansion for optical phase shifts is derived.

Quantum-mechanical scattering from an assembly of nonoverlapping potentials

Quantum-mechanical elastic scattering from an assembly of nonoverlapping potentials is treated in a $T$-matrix formalism in configuration space. For potentials satisfying a certain fairly weak separation condition, the total $T$ matrix for the assembly of potentials is expressed in terms of the $T$ matrices for the individual potentials. The relation to the equations in $N$-body theory is briefly discussed.

Model for scattering from the bound state of a two-body system

An exactly solvable model for scattering of a particle from the bound state of a two-body sysem is studied. In this model the motion of all particles is confined to one dimension. It is also assumed that the interaction between the projectile and the target is nonlocal and separable, but the potential between the target particles is local. Depending on the form of the latter potential two cases are considered: (1) the target is the ground state of a system whose eigenvalues are discrete and eigenfunctions localized, and (2) the target has only one bound state. In both cases the problem can be solved with overlapping as well as nonoverlapping potentials. This model is used to test the accuracy of the fixed scatterer approximation. It turns out that in this approximation the two cases yield similar results.NUCLEAR REACTIONS One-dimensional model. Separable forces. Fixed scatterer approximation.

Glauber theory and scattering off two centres

The Glauber additivity amplitude for scattering off two fixed centres is presented in a simple form for general arrangements of the centres. It is studied with respect to scattering off truncated Yukawa potentials. Scattering amplitudes are computed for three separations of the centres, corresponding to coincident, partially overlapping and non-overlapping potentials. The Glauber amplitude is compared with the impulse approximation for all separations, and with the exact solution for coincident and non-overlapping centres. It is also shown that the Glauber amplitude approximately satisfies the t-matrix unitarity condition for any arrangement of the two centres.

Corrections to the Glauber multiple scattering series

The asymptotic accuracy of the Glauber multiple scattering formula is examined in the fixed scatterer approximation in one dimension. If the potentials overlap, the off-shell corrections that dominate are asymptotically of the same order as the corrections to the eikonal approximation results. In the case of nonoverlapping potentials, the corrections to the Glauber formula arise from multiple scattering. The asymptotic behavior of these latter corrections depend essentially on the smoothness of the potentials.

Scattering from nonoverlapping potentials. III. Pion elastic scattering on 4He in the 3,3 region

The π − 4 He elastic scattering in the 3,3 resonance region is discussed in the framework of the nonoverlapping potentials model. Single and double scattering terms are considered together with their modification by reflection terms, which allow the projectile to oscillate back and forth an arbitrary number of times within a pair of nucleons. Reflections and spin and isospin flips are found to constitute a determining factor in a qualitative reproduction of the correct shape of observed angular distributions, in particular for backward scattering angles. Reflections are probably unimportant in πd scattering.

Scattering from nonoverlapping potentials II. Extension and comparison with glauber multiple scattering theory for deuterium

Scattering on a two-body nuclear target is considered in the nonoverlapping potentials model. The double scattering terms are discussed and it is suggested that their off-shell propagation part is not cancelled by higher-order terms of the multiple scattering series. The effect of reflections is discussed and practical ways to sum up their effect are offered. Finally, spin degrees of freedom are introduced within the nonoverlap formalism.

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate. Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio V E . Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.

Non-overlapping potentials and the Ericson-Ericson renormalization in the pion-nucleus optical potential

Abstract Do hadron-nucleus scattering amplitudes and properties of exotic atoms depend on the off-shell values of the elementary hadron-nucleon scattering amplitude? This problem is investigated within a multiple scattering formalism based on the frozen-nucleus approximation. If repulsive nucleon-nucleon correlations prevent the overlap of the various hadron-nucleon interaction regions, only on-shell values are needed. For this case, an expansion of the optical potential is proposed which preserves the on-shell property at each level of approximation. In lowest order, the Ericson-Ericson renormalization is recovered for the impulse approximation to the optical potential. Properties of the local and non-local forms of the π -nucleus optical potential are proven. Multiple scattering in the presence of the Coulomb potential is formulated and a kind of minimal-coupling prescription is derived.

Multiple Scattering by Two Centers of Force

A solvable model for scattering by two centers of force is studied. The force at each center is assumed to be a separable force acting on the S wave only. A force law of this kind enables one to find the scattering amplitude for overlapping as well as nonoverlapping potentials analytically. The resulting scattering amplitudes can be used to compare the range of validity of the Born and other high energy approximations and also to test the accuracy of the result obtained by retaining the leading terms in the multiple scattering series. By examining the relation between the exact scattering amplitude for the two centers and the scattering amplitude of one center, it is shown that in the case of nonoverlapping potentials, the former amplitude can be expressed completely in terms of the latter. Hence no additional information about the nature of force results from the determination of the two-center scattering amplitude. However, for overlapping potentials, one can discriminate among equivalent potentials by finding the scattering amplitude for the two-center-of-force problem. Finally to justify the use of a potential which acts only in one partial wave (l = 0), the following observation has been made. If one calculates the scattering length for two nonoverlapping hard sphere interactions, using S-wave scattering by each center, and then repeats the calculation for the same interactions but considers S and P waves and their interference, the difference between the two results will be quite small.

Short-Range Order and the Electronic Structure of Binary Alloys and Liquid Metals: A Comparison of Different Approaches

A number of exact results concerning the effects of short-range order on the energy spectrum of disordered systems are derived. In a single-band model of the substitutional alloy, it is shown that the height and width of the minority subband must scale as ${[\frac{x}{(1+\ensuremath{\epsilon})}]}^{\frac{1}{2}}$ and ${[x(1+\ensuremath{\epsilon})]}^{\frac{1}{2}}$, respectively. Here $x$ is the impurity concentration and $\ensuremath{\epsilon}$ is a parameter that specifies the degree of short-range order, i.e., the enhanced probability that impurity atoms will be nearest neighbors. These results depend only on the form of the (nearest-neighbor) two-site distribution function and it is, therefore, rather surprising that they are not reproduced exactly by any of the existing theories of short-range order. In each case, the difficulty can be traced to an improper factorization of the three-site distribution function. As a simple model of the liquid metal, we consider a close-packed array of nonoverlapping spherical-well potentials of depth $\ensuremath{\delta}$, radius ${R}_{m}$, and mean density $n$. This model is used to compare alternate formalisms: the quasicrystalline approximation of Lax, and the more recent approaches of Gyorffy, and of Schwartz and Ehrenreich. The fact that the short-range order guarantees that the potentials are nonoverlapping allows us to extract from each order in perturbation theory the leading contribution to the moments of the electron self-energy. The result is simply expressed in terms of the renormalized cumulants ${Q}^{(p)}(x)$, where $x$ is the dimensionless density parameter $x=\frac{n4\ensuremath{\pi}{{R}_{m}}^{3}}{3}$. These contributions isolate just those terms of higher order in $x$ that are most important for the moments of the electronic spectrum and are reproduced to all orders by only the equations of Schwartz and Ehrenreich.

Multiple Scattering and Nuclear Information

One-dimensional scattering from two mobile rectangular potential wells in a bound state is studied in order to investigate the contributions from the overlapping-potential region. The above model is used to compare the exact solution in the "fixed-scatterer" approximation with the corresponding ones obtained assuming non-overlapping potentials and additivity of phase shifts. The kinematics of the "fixed-scatterer" and "impulse" approximations are also examined, and the differences are found to be still significant at intermediate energies. For non-overlapping potentials the composite scattering is completely determined in terms of the individual scattering coefficients or on-shell information, and the solution is given by a linear system of coupled algebraic equations. Unfortunately, as far as scattering information is concerned the overlapping-potential region is the interesting one, since it offers the possibility of discriminating among phase-shift-equivalent potentials and of determining the short-range behavior of the wave function. Our results suggest that the predictions provided by the non-overlapping potentials or additivity of phase-shifts assumptions deviate sufficiently from the exact result to render ambiguous the extraction of the distribution function or two-body interaction from elastic intermediate-energy scattering in terms of either of these two models.