Atomic sphere approximations

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

The muffin-tin potential around each atom in the unit cell has been calculated in the framework of the Local-Spin-Density-Approximation using the ASW method. The ASW method uses the atomic sphere approximation (ASA), i.e. for each atom a sphere radius is chosen such that the sum of the volumes of all the overlapping spheres equals the unit cell volume. The calculation yields the expected ferromagnetic coupling between Cr and Ni. From the self-consistent spin polarized DOS, partial and total magnetic moment per formula unit can be computed. The calculated total magnetic moment is 5.2 pg in agreement with the experimental value (5.3 0.1 e calculations presented here have been performed... 

Antiphase boundary (APB) conservative vacancy segregation at Arrhenius plot Asymmetrical mixtures Atomic-sphere approximation (ASA) ASA-LSDA... 

The results are conveniently and clearly expressed in a thermodynamic formalism this is why they find their place in this chapter. They depend however on parameters which are drawn from band-theory, especially from the LMTO-ASA (Linear Muffin-Tin Orbitals-Atomic Sphere Approximation) method. 

Electronic structure determinations have been performed using the self-consistent LMTO method in the Atomic Sphere Approximation (ASA). 

Fig. 55. Debye temperature, d, and density of states at the Fermi level, N(Ep), for Y(Ni xCox)2B2C and Y(Ni xCux )2B2C as a function of the Co/Cu substitution level x. Symbols results derived from a relativistic band calculations in the atomic sphere approximation. Curves (in lower panel) rigid band model. After Ravindran...

TABLES 1 and 2 show the calculated and measured results of splitting energies in WZ and ZB structures, respectively. Suzuki et al derived the values of A and Ar for WZ and ZN GaN and AIN from a full-potential linearised augmented plane wave (FLAPW) and band calculation [3,4], Another result with LAPW calculation was given by Wei and Zunger [5], Kim et al [6] determined them by the linear muffm-tin orbital (LMTO) method within the atomic sphere approximation (ASA). Majewski... 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Duthie and Pettifor (1977) have treated d bands and, s bands in the rare earths and have considered the structure dependence by calculating the density of states in detail (within the Atomic Sphere Approximation described in Section 20-D) for different structures. They indicate that they have predicted a sequence of four different structures, which occurs both for increasing pressure and for decreasing atomic number across the rare-earth series. This correlation and the ratio of core volume to atomic volume had been related by Johansson and Rosengren (1975), but Duthie and Pettifor argue that the essential feature is the number of electrons in the d bands and that this is only incidcntly rcllected in the core volume. 

We have used the Atomic Sphere Approximation only to obtain values for and Ej. The approach can be used as the basis for more complete calculations, as Pettifor (1977a,b) did in his detailed study of the 4r/ series. 

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

Now let us develop the relation between F and the band width. The derivation is due to Heine (1967), who applies the same boundary conditions at the atomic sphere that we described for the Atomic Sphere Approximation. The bottom of the (I band was determined by setting the gradient of the cl state for each atom equal to zero at its atomic sphere, or... 

Interestingly enough, it has been possible to write the free-electron effective mass in terms of the same r/-state radius that determined the d-band width and determined hybridization with the free electrons. It was noted in the discussion of the Atomic Sphere Approximation that Andersen defined an effective mass for d electrons in terms of the r/-band width, = 12.5fiV(m, ro) This can be combined with the expression for in terms of (Eq. 20-9), to write m /m = (1 + 2.9lm/m ). In the Atomic Sphere Approximation Wp, and arc regarded as independent quantities, but both the and values given by Andersen and Jcpsen (1977) are rather close to the effective mass m., obtained from m,/m = (I + 2.91j /ni,j) . ... 

---