Quasi-crystalline Approximation Research Articles

Muffin-tin model of a liquid metal: Electronic density of states in a modified quasicrystalline approximation

In the muffin-tin model of a liquid metal, the individual scattering centers are represented by nonoverlapping, spherically symmetric atomic potentials. Given a specific configuration of the scatterers, it can be shown that the atomic potentials enter the calculation of the density of states $\ensuremath{\rho}(E)$ only through the atomic phase shifts ${\ensuremath{\delta}}_{l}(E)$. The quantities of physical interest, however, are given by an average over all possible configurations and are usually evaluated in terms of a single site decoupling of the multiple-scattering series. Once such an approximation is made, it is no longer clear that the mean density of states $〈\ensuremath{\rho}(E)〉$ can be expressed in terms of just the quantities ${\ensuremath{\delta}}_{l}(E)$. [Physically, the decoupling may introduce a spurious overlap of the potentials and the evaluation of $〈\ensuremath{\rho}(E)〉$ would then require the off-shell matrix elements of the atomic-scattering operators.] In particular, the familiar quasicrystalline approximation (QCA) is shown to exhibit this deficiency. We suggest a modified QCA which overcomes this deficiency by providing a more realistic description of the atomic correlations. The applicability of our method is illustrated by calculations based on a one-dimensional model. We also discuss the qualitative behavior of the quasiparticle spectrum of a realistic three-dimensional transition- or noble-metal liquid.

Short-range order and the electronic structure of a one-dimensional liquid metal

The energy spectrum of a one-dimensional array of $\ensuremath{\delta}$-function potentials is studied as a function of the degree of short-range order in the arrangement of the scatterers. The integrated density of states is evaluated numerically and the results are used to compare alternate theoretical formalisms. In particular, we consider Lax's quasicrystalline approximation (QCA) and the more recent self-consistent approaches of Gyorffy and Schwartz and Ehrenreich. Surprisingly, we find that only the non-self-consistent QCA yields a reasonable description of the electronic spectrum. The failure of the self-consistent schemes appears to be due to an incomplete treatment of the multiple-occupancy corrections.

Phonons in disordered systems with strong short-range order

A theoretical study is made of phonons in cellular disordered systems by using an approximate averaged equation which corresponds to a slightly generalized quasi-crystalline approximation. This method allows the correlation between neighbouring particles to be taken into account. It is concluded that the exact solutions are obtained in the limiting cases where the correlation forces the system to be an aggregation of ordered systems. Numerical calculations of the state density and the local state density are made for the one-dimensional case over a wide range of the degree of correlation. These calculations show that the method not only can predict, to a considerable extent, the peak structure which characterizes the cellular disordered systems, but also can predict some local features of phonons in these systems.

On the CPA conductivity of randomly disordered systems

It is shown that the decoupling assumption as usually adopted in the coherent-potential approximation approach for randomly disordered alloys is equivalent to some quasicrystalline approximation. An extension to random structurally disordered materials is straightforward, but if some short-range order is taken into account serious difficulties are encountered. Finally, a sufficient condition is derived for the atomic vertex correction to be zero, starting from an analysis for a δ-function potential model.

Tight-Binding Models of Amorphous Systems: Liquid Metals

A tight-binding approach to the electronic structure of disordered systems is developed for a simple one-orbital model of a liquid metal. An equation is derived for a one-electron continuum Green's function from which the electronic density of states can be obtained. Utilizing an analogy between this Green's function and the $T$ matrix of multiple-scattering theory, results are obtained corresponding to the quasicrystalline approximation (QCA) of Lax and the self-consistent approximation (SCA) of Schwartz and Ehrenreich. Moments of the spectral function are also analyzed. Calculations were made using random and hard-sphere pair distribution functions. The QCA in this model is quite inadequate, and the SCA, while a considerable improvement, proves to involve a questionable approximation to the three-body distribution function.

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.

On a Problem in Random-Lattice Statics

The quenching of fluorescense in pure solid phosphors due to the well-known mechanism involving the resonance transfer of the excitation energy from one luminescent centre to another and eventually to randomly distributed imperfections acting as sinks, or traps, is reformulated as a problem in random-lattice statics. Here the question of configurational averaging over the random sinks, involvied in the calculation of the fluorescence yield, or efficiency, has been treated in an approximation that goes beyond the virtual-crystal approximation (VCA), and presumably incorporates the quasicrystalline approximation (QCA). An expression has been derived for the efficiency of fluorescence (η) in terms of the trap concentration C t and the phenomenological rate constants 1/τ, Γ and χ characterizing, respectively, the de-excitation of the luminescent centre by re-radiation, resonance transfer to another luminescent centre, and through capture by a trap. This has been done for the case of an idealized phosphor consisting of a lattice of luminescent centres with randomly substituted traps, assuming nearest-neighbour transfers only. It is found that η falls off rapidly with increasing Γ in qualitative agreement with some recent experiments on the extinction of luminescence in certain weakly exchange-coupled paramagnetic phosphors.

On the application of the Van Hove theory to disordered systems

It is pointed out that Van Hove's theory as applied to the random-lattice problem is equivalent to the wrong Σ (1) approximation of the Green-function diagrammatic method. It is also indicated that van Hove's theory together with the “geometric series” approximation applied to the theory of electron states of liquid metals is equivalent to the simple generalization of so-called quasi-crystalline approximation.

Canonical density matrix for muffin tin potentials. (Liquid metals)

A direct method for constructing the canonical density matrix C(r, ro; beta ) for a single site problem using the complete set of normalized scattering solutions of the Schrodinger equation is described. For a spherical potential well the partial wave density matrices, Ci(r, r0, beta ), bring out several qualitative features. By this construction, it is shown that the information put into the many site problem of a liquid are on the same footing as in other single site approximations, namely, (i) phase shifts, and (ii) the radial distribution function. Since the Laplace transform of C(r, r0; beta ) is related to the Green function G(r, r0 -E), the usual Green functions of the conventional scattering theory can be used to compute C for simple systems. Finally, it is shown that the method of Rousseau, Stoddart and March (1970) for calculating the density of electronic levels in liquid metals is a single site approximation and is similar to the quasicrystalline approximation given by Ziman (1966).

On the electronic structure of disordered alloys. I

The average Green function and the self-energy of a many-component disordered alloy is determined in a quasi-crystalline approximation. Second-order correlations between different lattice sites are taken into account. The summation method of Lloyd for the calculation of the density of states for a liquid is modified to make it applicable for an alloy. Finally the coherent-potential approximation is discussed for the considered general alloy model.

A formal theory of conductivity in disordered materials

A formal approach is established in order to calculate the electrical-conductivity tensor for disordered materials. Binary and more general alloys as well as liquid metals are considered. Throughout this paper a quasi-crystalline approximation is used to evaluate the appropriate configurational averages. The value of such an approximation is also discussed in some detail.

Single-site approximations in the electronic theory of liquid metals

The electronic structure of liquid metals is described by a multiple scattering formalism. Alternate methods based on a single site description of the liquid are developed and compared. The first method, the quasicrystalline approximation of Lax, is used to interpret and extend the work of Ziman and Anderson and McMillan. The second method, discussed here for the first time, extends the coherent potential theory of binary substitutional alloys to the liquid problem. Formally, this method is based on the introduction of a local self-energy operator associated with each site, and the subsequent self-consistent decoupling of the multiple scattering hierarchy. The lowest-order effects of one and two atom clusters are then discussed in connection with the expansion of the total scattering operator. It is shown that in the liquid, as in the alloy, these effects are treated more accurately by the self-consistent methods than by their non-self-consistent counterparts, the quasicrystalline and average t matrix approximations. It is in fact possible to establish a parallel between the self-consistent and coherent potential approximations on the one hand and the quasicrystalline and average t matrix approximations on the other. In the evaluation of second-order corrections to the work of Anderson and McMillan, unitarity requirements suggest an expansion in terms of the reaction matrix rather than the scattering matrix. This procedure treats the damping of electrons more carefully, eliminating spurious contributions. These formal considerations are illustrated by a calculation of the complex band structure and density of states for liquid Cu.

A Theory of Phonons in Amorphous Solids and Its Implications to Collective Motion in Simple Liquids

A theoretical study is made of phonons in amorphous solids using a self-consistent phonon scheme. In treating the structural disorder inherent in the problem, a theory is formulated from the viewpoint of multiple scattering theory and designed to give phonon eigen-frequencies which are expressed in terms of many-body correlation functions of atoms as well as of interatomic potentials in the solids. For this purpose, a conditional averaging procedure is applied to equations for phonon Green's functions, obtainable using the renormalized harmonic approximation. A set of hierarchy equations is thereby obtained, for which various decoupling approximations are employed. As an application of the results obtained here, numerical calculations of the eigenfrequencies of longitudinal and transverse phonons in liquid argon are using the “quasi-crystalline approximation” and the Lennard-Jones model potential. Fairly good agreement with experiment is obtained. A systematic and approximate self-consistent method for treating the hierarchy equations, which may be of some use in studying the general properties of the energy spectra of elementary excitations in disordered systems, is also obtained.

Electronic structure of disordered materials with strong angular correlations

Introducing the concept of a paracrystal, a dispersion relation is obtained relating the energy to the wave vector of an electron, moving in a disordered structure with strong angular correlations. The potential field is assumed to be of the muffin-tin type and concerning the configurational averaging technique, the quasi-crystalline approximation is used throughout the paper. The results of the KKR method for the study of energy bands in crystals and of the PZ theory concerning the electronic structure of liquid metals are particular cases of the general theory. Finally, the method is applied to a simple model of an assembly of pointscatterers, constituting an ideal paracrystal.

Electronic States in Liquid Metals: A Generalization of the Coherent-Potential Approximation for a System with Short-Range Order

The problem of finding the ensemble averaged Green's function $〈G〉$ which describes an electron moving in the potential field of $N$ stationary scatterers is formulated in terms of multiple scattering theory. The correlations between the positions of the scatterers are explicitly taken into account. By making use of the quasicrystalline approximation and a generalization of the coherent-potential prescription, a procedure for calculating $〈G〉$ is proposed. Because of the use of the quasicrystalline approximation, it involves only the radial distribution function of the scatterers in addition to the scatterer-electron interaction potential. It is shown that the procedure reduces to the coherent-potential approximation for randomly distributed scatterers, i.e., no short-range order, and to Korringa's band-structure calculation for an order lattice. It is pointed out that the method is applicable to the problem of electronic states in random alloys and that of vibrational spectra of imperfect crystals.

On the LCAO method for disordered materials: I. General Theory

The purpose of the present paper is to show how the LCAO method, well-known in crystal physics, can be generalized to disordered structures. Using a quasi-crystalline approximation, we obtain an equation relating the total energy of an electron in the considered medium to its wave vector. This relation depends on the pair correlation function or the radial distribution function of the system and defines the electronic band structure of the material. The approach discussed in the second section, can also be generalized to include degeneracy of the atomic energy levels as well as atomic energy level interactions.