Theoretical Approach for Long-Ranged Local Lattice Distortion in Al-Rich AlX (X = H∼Sn) Disordered Alloys by Kanzaki Model Combined with KKR Green’s Function Method

Abstract

The local lattice distortion (LLD) effects in the disordered alloys with the large atomic-size misfit between the constituent elements are well known to be essential for reproducing theoretically the observed phase diagrams including order-disorder critical temperatures, although the theoretical approach from first principles remains a long-standing problem. We propose the Kanzaki model combined with the full potential (FP) Korringa-Kohn-Rostoker (KKR) Green’s function method, as an approximation for the direct FPKKR calculations, which is based on the harmonic approximation of the atomic displacements and enables us to study the long-ranged LLD effects in the disordered alloys. We first show that the present Kanzaki model may reproduce very accurately the LLD energies obtained by the direct FPKKR calculations with the restriction on the displacement of only the 1st-nearest neighboring atoms around a single impurity X in Al, although the discrepancy increases with the atomic-size misfit between Al and X atoms. Second we clarify the fundamental features of the LLD energies (over ∼10000 atoms) around a single impurity X (= H∼Sn) in Al, corresponding to the 1-body part in our real space cluster expansion for the LLD energy in the Al-rich AlX disordered alloy; the 1-body LLD energy becomes larger and more long-ranged with the atomic-size misfit between Al and X elements and also with the sp-d interaction of Al-X (X = d element).

Fig. 5 Atomic displacements of the host atoms in the neighborhood of a single impurity X in Al, up to the 20th-nn, obtained by the Kanzaki model combined with the GGA-FPKKR method: (a) X = Mg, (b) X = Sc, (c) X = Cu, (d) X = Zn, (e) X = Rb, (f) X = Zr, and (g) X = Ru. There are two equivalent sites for the 9th-, 13th-, 16th-, 17th-, and 18th-nn host atoms. The HF-forces up to the 10th-nn host atoms (in Figs. 4) are also shown in order to compare with the atomic displacements. The numerical scale for the HF forces is the same as that for the atomic displacements (the left-hand side).

1. Introduction

The presence of point defects in a crystal, such as vacancies and impurity atoms, generally causes displacements of the neighboring host atoms from their ideal lattice positions. For alloys, such a local lattice distortion (LLD) changes the lattice constant and this change can be measured by x-ray diffraction. King calculated quantitative size-factors for 469 substitutional solid solutions, defined in terms of the effective atomic volume of the impurity atoms by using the experimentally-known precise lattice parameters available in the literature.¹⁾ Values of the volume size-factor, its linear derivative and a parameter expressing the deviation from Vegard’s law, were tabulated in Ref. 1). However, this information for the volume relaxations of the A-rich AX alloys is not sufficient to estimate the interatomic distances around the solute atom X because the LLD is generally different in magnitude for the different atomic shells around the defect. More detailed information can be obtained by extended x-ray-absorption fine-structure (EXAFS) experiments. A systematic study for the LLD effects around substitutional impurity atoms in fcc A-based (A = Al, Cu, Ni, Pd, and Ag) and bcc B-based (B = Fe, Nb, and V) dilute alloys was reported by Scheuer and Lengler, who determined interatomic distances, coordination numbers, and Debye-Waller factors.²⁾

On the other hand, the theoretical investigation for the LLD effects around defects in crystals is a difficult task. In the past, this problem was mostly dealt with on a phenomenological basis,^3,4) e.g., by applying models of lattice statics or continuum theory. Various semiempirical methods have also been employed, especially for defects in semiconductors.⁵⁾ Since a reliable microscopic description of the LLD energies based on ab-initio electronic structure calculations requires very accurate total energies and Hellman-Feynman (HF) forces, the ab-initio calculations for the LLD energies have mostly been attempted so far for simple metals and semiconductors on the basis of pseudopotential treatment.^6–9)

The difficulty in calculating the LLD energies, defined by the total-energy difference between the ideal and equilibrium atomic positions, arises mainly from the fact that the LLD effects around X impurities in A metal may become larger and more long-ranged with the atomic-size misfit between A and X elements, as shown quantitatively in the present paper. For example, the LLD energy becomes as small as ∼−0.001 eV for a single impurity X = Zn in Al (the impurity system with the very small atomic-size misfit among X = H∼Sn in Al), and as large as ∼−0.05 and ∼−0.07 eV for X = Mg and Cu in Al (the impurity systems with the relatively large atomic-size misfit), and as very large as ∼−1.1 eV for X = Rb in Al (the impurity system with the largest atomic-size misfit among X = H∼Sn in Al). The LLD effects at the atomic sites far away from a single impurity X in Al also increase very much with the atomic-size misfit (see Sect. 3.4). It is also noted that the LLD energy may become larger with the aggregation of impurities, resulting in the phase separation, as discussed later for the Al-rich AlX (X = Mg, Cu) alloys.

The Jülich group and our group successfully applied ab-initio calculations based on the full-potential (FP) Korringa-Kohn-Rostoker (KKR) Green’s function method to the LLD energies around single impurities or two impurities in metals,^10–15) although the atomic displacements were restricted in the vicinity of single impurities or two impurities. The available experimental results for the atomic displacements in the vicinity of a single impurity X (= Sc∼Ge, Zr∼Sn) in Cu, Al and Fe, being listed in Ref. 2), have been reproduced accurately by the FPKKR calculations. The experimental results for the volume relaxations (an expansion or a compression from the atomic volume of A metal) of A-rich AX (A = Al, Fe, X = Sc–Ge) alloys, being listed in Ref. 1), have also been reproduced very accurately by using the simple formula for the volume change due to a single impurity X in A (= the 1st-moment of the Kanazki forces around a single impurity X in A metal, given by eq. (31) in Ref. 10)).^3,10,11,13)

We are now developing an ab-initio calculation method to study the thermodynamic properties of the ordered and disordered alloys. We have already succeeded in quantitatively reproducing the X-concentration dependence of the observed high solvus temperatures (= 800∼1600 K) of Pd-rich PdX (X = Ru, Rh) alloys by the free energy calculations based on the real space cluster expansion (RSCE) for the internal energies of alloys, from a dilute limit (as an impurity problem), and the cluster variation method in the tetrahedron approximation (CVMT) and the tetrahedron-octahedron approximation (CVMTO) for the configurational entropy.^16–19) The n-body (n = 1∼4) interaction energy (IE) among X impurities in Pd, used in the RSCE for the internal energies of Pd-rich PdX alloys, is divided into the chemical interaction (CI) energy at the ideal atomic positions and the LLD energy due to the atomic displacement around X impurities in Pd, both of which can be uniquely and successively determined from the 1-body to 4-body by the ab-initio calculations based on the FPKKR Green’s function method for the perfect and impurity systems (Pd-host and X_n in Pd, n = 1∼4), combined with the generalized gradient approximation (GGA) in the density functional theory (DFT).^18,19) We investigated the characteristic features for the CI part in the n-body IE among X (= Ru, Rh) impurities in Pd: (I) the fundamental features of the 2-body IEs, including the distance dependence up to the 20th-nearest neighboring (nn); (II) the thermal electronic contribution due to the Fermi-Dirac (FD) distribution. It is also noted that the global volume relaxations dependent on both of X-concentration and temperature can be treated routinely and simply by our approach.^16,17,20,21) We also calculated the LLD energies in the 1st-nn 2-body IEs, being the deviation from the global volume relaxations and existing even at 0 K, although the LLD effects were restricted in the vicinity of X impurities. At present, it is very difficult for the FPKKR calculations to take into account the LLD effects at the atomic sites far away from X impurities.

In order to study the phase diagrams of the age-hardenable Al-rich AlX alloys such as X = Mg, Cu, and Zn, we are now calculating the n-body IEs among X impurities in Al, following the above-mentioned approach.^16–19) We have already calculated the CI parts in the n-body IEs for X (= H∼Sn) in Al and clarified the characteristic features of the above-mentioned effects (I) and (II).²²⁾ The LLD effects, being the deviation from the global volume relaxations, seem to be also very important for the Al-rich AlX disordered alloys. Differently from Pd-rich PdX (X = Ru, Rh) disordered alloys,^16,17) for example, the LLD effects in the Al-rich AlX (X = Cu, Mg) disordered alloys may be large and long-ranged because Al is a soft metal²³⁾ and the atomic-size misfit between Al and X (= Mg, Cu) elements is relatively large (∼11 and ∼10%).²⁴⁾ We have already found that the CI parts in the 2-body IEs for X (= Mg, Cu) in Al are positive (repulsion),²²⁾ so that a simple consideration of the CI contribution alone contradicts completely the experimental results (segregation at low temperatures) for the phase diagrams of the Al-rich AlX alloys.

The importance of the LLD effects in the disordered alloys with the large atomic-size misfit between constituent elements was already indicated out by Wei et al.²⁵⁾ and Sanchez et al.²⁶⁾ According to the work of Sanchez et al.,²⁶⁾ the IEs of AgCu alloys with the relatively large atomic-size misfit (∼11%)²⁴⁾ changes drastically (for example, from repulsion to attraction of the effective 1st-nn 2-body IE) by including the contribution due to the local volume relaxations, resulting in good agreement with the observed phase diagram. They calculated the n-body IEs by using total energies at each equilibrium volumes of the five ordered phases of AgCu alloys (Ag_4−nCu_n, n = 0∼4). Their results mean that the n-dependent local volume relaxations are very important to reproduce the observed phase diagram of the AgCu alloys.

However, the solvus temperatures of the Ag-rich AgCu alloys are fairy overestimated with the Cu-concentration, as shown in Fig. 5 in Ref. 26), due to the simple treatment of Sanchez et al. The five equilibrium volumes of AgCu alloys (Ag_4−nCu_n, n = 0∼4) may be obviously much smaller (with n) than those of the Ag_4−nCu_n in the Ag-rich AgCu alloys, because the atomic radius of Cu is much smaller (∼11%) than that of Ag.²⁴⁾ Thus, the overestimation for the solvus temperatures of the Ag-rich AgCu alloys may be due to the overestimation for the attractive effects of the IEs (segregation effects of Cu atoms), caused by the overestimation of the local volume relaxations.

On the other hand, in order to examine the importance of the LLD effects (for the n-body IE, n = 0∼4) in the AuCu alloys with the relatively large atomic-size misfit (∼11%) between the constituent elements,²⁴⁾ Wei et al. assumed an appropriate model with five adjustable parameters for the local volume relaxation effects of the five tetrahedra (Au_4−nCu_n, n = 0∼4) in the Au_1−cCu_c disordered alloy, which treats the dependences both on c and n for the local volume relaxations of Au_4−nCu_n in the Au_1−cCu_c disordered alloy, as the deviation from the equilibrium local volumes of the Au_4−nCu_n (n = 0∼4) ordered alloy.²⁵⁾ As shown in Fig. 13 in Ref. 25), they obtained the nearly perfect fit with the observed phase diagram for the AuCu alloys, by choosing appropriately a single effective value for five adjustable parameters (n = 0∼4) and concluded that the LLD effects in the disordered AuCu alloys with the relatively large atomic-size misfit between the constituent elements are essential for reproducing the observed order-disorder critical temperatures.

Thus, the accurate treatment for the LLD effects in the disordered alloys, based on the ab-initio calculation method, is strongly requested in order to investigate theoretically the thermodynamic properties of the disordered alloys with the large atomic-size misfit between the constituent elements. As mentioned above, we have already succeeded in reproducing very accurately the observed volume relaxations of the A-rich AX (A = Al, Fe; X = Sc–Ge) disordered alloy by using the simple formula with the Kanzaki forces at the A atoms (at the atomic positions up to the 10th-nn) around a single impurity X in A.^11,13)

The purposes of the present paper are: (1) to propose the Kanzaki model combined with the GGA-FPKKR method for the impurity systems, as an approximation for the direct GGA-FPKKR calculations, which is based on the harmonic approximation^3,10,27) and enables us to calculate the long-ranged LLD energies around X_n impurities in Al; (2) to apply the model to the calculations for the LLD energies (for example, over ∼10000 atoms) around a single impurity X in Al, corresponding to the LLD parts in the 1-body IEs for the internal energies of the Al-rich AlX alloys;^16–18,22) (3) to clarify the fundamental features of the long-ranged LLD energies around a single impurity X in Al.

In Sect. 2, we discuss the FPKKR method for total energies and HF-forces, and the present Kanzaki model for the LLD energies around X impurities in Al. Section 3 shows the calculated results for a single impurity X (= H∼Sn) in Al. In Sect. 3.1, we show that the chemical trend of the equilibrium lattice parameters of X (= H∼Sn) and Al₃X metals in fcc structure can be understood systematically by the sign and magnitude of HF-forces at the 1st-nn atoms around a single impurity X in Al. In Sect. 3.2, in order to examine the accuracy of the HF-force obtained by the FPKKR calculations, we show that the atomic displacement of the 1st-nn atoms, determined from the zero-force condition, agrees very well with that determined from the minimum of the total energy obtained by the FPKKR calculations, for X = Mg, Cu, Zn, Sc, Zr, Ru, and Rb in Al. The elements of X = Mg, Cu, and Zn were selected as the important solute elements for the age-hardenable Al-rich AlX alloys, while X = Sc and Zr were selected as the representative elements of 3d and 4d element series, which are recently known to be very attractive because Sc and Zr added Al-based alloys exhibit excellent mechanical properties.²⁸⁾ X = Ru was selected as the element of the strongest sp-d interaction (of Al-X) among X = 3d and 4d impurities,²²⁾ while X = Rb as the element of the largest atomic-size misfit among X = H∼Sn in Al. In Sect. 3.3, in order to examine the accuracy of the present Kanzaki model, we compare the calculated results from both of the direct GGA-FPKKR method and the prsent Kanzaki model. We show that the present Kanzaki model calculations may reproduce very accurately the LLD energies obtained by the direct FPKKR calculations, although the discrepancy between them becomes larger with the atomic-size misfit between Al and X elements. In Sect. 3.4, we clarify the fundamental features of the long-ranged LLD energies (over ∼10000 atoms) around a single impurity X in Al. We show that the LLD effects around a single impurity X in Al become larger and more long-ranged with the atomic-size misfit between Al and X elements, and also with the sp-d interaction of Al-X (X = d element), as shown for the CI part in the 2-body IE.²²⁾ Section 4 summarizes the main result of the present paper and discusses the future problem for the LLD parts in the n-body (n = 2∼4) IEs for the internal energies of the Al-rich AlX (X = H∼Sn) disordered alloys.

2. Calculation Method

2.1 Total energies and HF-forces of the impurity systems

The calculations for the total energies and HF-forces of the impurity systems are based on the DFT in the GGA.^10,23,29) In order to solve the Kohn-Sham equations, we use a multiple scattering theory in the form of the FPKKR Green’s function method. We use the screened version of the FPKKR band calculations for the Al-host, which significantly simplifies the numerical calculations by introducing the short-range structural Green’s functions.^30,31) In order to simplify the total-energy calculations in the GGA formalism, we may use the electronic densities obtained self-consistently by the local spin density approximation (LSDA) in the DFT. This perturbative treatment is very useful for the total-energy calculations because we can avoid the slowly converging self-consistent iterations for the GGA Kohn-Sham equations. The accuracy of the non-self-consistent (NS) GGA calculations for a single impurity X in Al will be discussed in Sect. 3.2, while the accuracy of the NSGGA calculations for crystals was already shown in Ref. 29). The characteristic features and important advantages of the FPKKR Green’s function method, different from the usual supercell and cluster calculations, are discussed in Refs. 10), 22) and 23).

We now explain the RSCE for the internal energies of the Al-rich AlX alloys, from a dilute limit (as an impurity problem). The n-body (n = 1∼4) IE among X impurities is divided into the chemical interaction (CI) energy at the ideal atomic positions and the LLD energy due to the atomic displacements caused by the insertion of impurities. The CI parts in the n-body (n = 1∼4) IEs among the 1st-nn X impurities were already calculated uniquely and successively from the 1-body to 4-body, by the combination (like eqs. (6)–(9) in Ref. 18)) of the total energies of the 1st-nn tetrahedron impurity clusters (Al_4−nX_n, n = 0∼4) in Al.²²⁾ The LLD parts in the n-body IEs can also be calculated uniquely and successively from the 1-body to 4-body by the combination (like eqs. (6)–(9) in Ref. 18)) of the n-dependent LLD energies around X_n impurities, although the potentials in the impurity region up to the 1st-nn atoms around the displaced host atoms must be recalculated self-consistently because the potentials of the displaced atoms are strongly perturbed.^11–13) In order to calculate the LLD energy around a single impurity X in Al, we redetermine self-consistently all the potentials in the impurity region including 201 atoms up to the 10th-nn around X impurities in Al (see Fig. 1 in Ref. 13)).

2.2 Long-ranged LLD energy by Kanzaki model combined with GGA-FPKKR method

In order to calculate the long-ranged LLD energies in the Al-rich AlX disordered alloys, we use the Kanzaki model combined with the GGA-FPKKR method, as a simple approximation for the direct GGA-FPKKR calculations. The Kanzaki model is based on the harmonic approximation for the atomic displacements^3,27) and may be expected to be a good approximation, at least, for the alloys with the small atomic-size misfit between the constituent elements. In the harmonic approximation for the atomic displacements, using the HF-force (Fⁿ) and the atomic displacement (s_n) at the site n in the defect system, the elastic energy is written as follows,   

\begin{equation} E_{\textit{elastic}} = \frac{1}{2} \sum\nolimits_{n,n'} \boldsymbol{s}_{n} \Phi^{nn'} \boldsymbol{s}_{n'} - \sum\nolimits_{n} \boldsymbol{F}^{n}\cdot \boldsymbol{s}_{n} \end{equation} (1)

where Φⁿⁿ′ is the force-constant matrix between the sites n and n′ in the defect system. Kanzaki developed a new formula to calculate E_elastic, by introducing the so-called Kanzaki forces $\boldsymbol{F}_{K}^{n}$, as follows,   

\begin{equation} E_{\textit{elastic}} = \frac{1}{2} \sum\nolimits_{n,n'} \boldsymbol{s}_{n} \Phi_{0}^{nn'} \boldsymbol{s}_{n'} - \sum\nolimits_{n} \boldsymbol{F}_{K}^{n} \cdot \boldsymbol{s}_{n} \end{equation} (2)

where $\Phi _{0}^{nn' }$ is the force-constant matrix between the sites n and n′ in the host crystal. Thus, the Kanzaki forces are defined as the forces that would induce the same local distortion in the host crystal as the direct forces (HF-force) cause in the defect system and include all the change of the force-constants due to the insertion of defects, as explained in the next paragraph. The schematic representation for the Kanzaki model is shown in Fig. 1.

Fig. 1

(a) Displacement field (s_n) in the defect system, caused by a substitutional defect such as a vacancy or an impurity, (b) the same displacement s_n, but produced by the Kanzaki forces $\boldsymbol{F}_{K}^{n}$ in the host crystal.

At the equilibrium positions of the atomic displacements, the s-derivatives of elastic energies of eq. (1) and (2) become zero. Thus, we can get the following relations, using the equilibrium values of the atomic displacements,   

\begin{equation} \boldsymbol{F}^{n} = \sum\nolimits_{n'} \Phi^{nn'} \boldsymbol{s}_{n'} \end{equation} (3)

  

\begin{equation} \boldsymbol{F}_{K}^{n} = \sum\nolimits_{n'} \Phi_{0}^{nn'} \boldsymbol{s}_{n'} \end{equation} (4)

Using eqs. (3) and (4), we can get the following formula for the Kanzaki forces,   

\begin{align} \boldsymbol{F}_{K}^{n} & = \sum\nolimits_{n'} \Phi_{0}^{nn'} \boldsymbol{s}_{n'} = \sum\nolimits_{n'} \Phi_{0}^{nn'}\boldsymbol{s}_{n'} + \left( \boldsymbol{F}^{n} - \sum\nolimits_{n'} \Phi^{nn'} \boldsymbol{s}_{n'} \right)\\ & = \boldsymbol{F}^{n} - \sum\nolimits_{n'}\varDelta \Phi^{nn'} \boldsymbol{s}_{n'} \end{align} (5)

It is noted that we must determine the Kanzaki forces self-consistently by using eqs. (2) and (5) since the Kanzaki forces ($\boldsymbol{F}_{K}^{n}$) depend on s_n′,

It is obvious that the accuracy of the Kanzaki model for the LLD energy depends on the accuracy of the input parameters ($\Phi _{0}^{nn' }$, Fⁿ and ΔΦⁿⁿ′) in the Kanzaki model. In the present approach, the Fⁿ and ΔΦⁿⁿ′, induced by the insertion of an impurity, can be calculated accurately by the GGA-FPKKR method for the impurity system at the ideal atomic positions. It is noted that ΔΦⁿⁿ′ is the difference between the force-constant matrices with and without a single impurity X, given by the difference between the slopes of the force curves.¹⁰⁾ It is also noted that, in order to increase the calculation accuracy for the HF-forces by the FPKKR method, we use the ionic HF-formula developed by the Jülich group.¹⁰⁾

On the other hand, the force-constant matrix Φ₀ of the Al host can be calculated from the present FPKKR method for the impurity systems, or one can use the Born-von Kármán parameters fitted to the experimental phonon dispersion curves. The accurate Born-von Kármán parameters for many metals were already tabulated in alphabetic order, by Dederichs et al.³²⁾ In the present study, for simplicity, we use the Born-von Kármán parameters fitted to the experimental phonon dispersion curves. It is well known that the Born-von Kármán parameters for Al metal are long-ranged and needed up to the 8th neighbors, in order to get the overall nice agreement with the experimental phonon dispersion curves.³²⁾ Thus, in the present work, we take into account the elastic energy up to the 8th neighbors around the region of the atoms displaced by a single X impurity in Al.

Lastly we discuss how to investigate the long-ranged character of the atomic displacements and LLD energies, caused by the insertion of a single impurity X in Al. We consider the atomic displacements of all the atoms in the cubic cell centered at the origin, with the length of one edge = na (a = lattice parameter). The coordinates of the Al atoms to be displaced are expressed as a/2(x, y, z) (−n ≤ x, y, z ≤ n, x, y, z = integer, and x + y + z = even number). We change n from 1 to 13 to investigate the convergency of the LLD effects with the distance, as discussed in Sect. 3.4. As a result, the total number (N) of the displaced Al atoms in the cubic cell changes from 12 to 9840, corresponding to n = 1∼13.

On the other hand, in the present Kanzaki model, the HF-forces are taken into account, at maximum, up to the 10th-nn (see Fig. 1 in Ref. 13)) around a single impurity X in Al, depending on the cubic-cell size (n). For example, for the cubic cell of n = 2, the HF-forces up to the 4th-nn (a/2(2, 2, 0)) are taken into account and, for the cubic cell of n ≥ 4, all the HF-forces up to the 10th-nn (a/2(4, 2, 0)) are taken into account. As discussed in Sect. 3.3, the HF-forces at the distant neighbors beyond the 10th-nn around a single impurity X in Al may be very weak. In the present Kanzaki model, for simplicity, we also assume in eq. (5) that only the force-constants between the impurity and its 1st-nn atoms change, as have been done in Ref. 10). This approximation may be not so bad because the force-constants between the distant neighbors beyond the 1st-nn sites are usually smaller by one order of magnitude than those between the 1st-nn atomic sites, for fcc metals such as Al, Ag, Cu, Ni, and Pd.³²⁾ It is also noted that the atomic displacements at the distant neighbors beyond the 1st-nn site are usually smaller than that at the 1st-nn site, as shown in Sec. 3.4.

3. Calculated Results

In Sect. 3.1, we show that the chemical trend in the variation of the equilibrium lattice parameters of X and Al₃X (= H∼Sn) metals in fcc structure, related to the equilibrium lattice parameter of Al metal, can be understood systematically by the sign and magnitude of the HF-force at the 1st-nn atom around a single impurity X in Al. In Sect. 3.2, in order to examine the accuracy of the HF-forces obtained by the all-electron FPKKR calculations with the ionic HF formula,¹⁰⁾ we show that the atomic displacement of the 1st-nn atom, determined from the zero-force condition, agrees very well with that determined from the minimum of the total energy. We show the calculated results for X = Mg, Cu, Zn, Sc, Zr, Ru, and Rb in Al, as the representative elements among X = H∼Sn, and discuss the accuracy of the NSGGA calculations, together with those of the self-consistent FPKKR calculations based on the LSDA and GGA. In Sect. 3.3, we show that the Kanzaki model combined with the FPKKR method may reproduce very accurately the LLD energies obtained directly by the FPKKR calculations with the restriction on the displacement of only the 1st-nn atoms around a single impurity X in Al, although the discrepancy increases with the atomic-size misfit between A and X atoms. In Sect. 3.4, using the present Kanzaki model, we clarify the fundamental features of the long-ranged LLD effects (over ∼10000 atoms) around a single impurity X in Al.

In the present work, for simplicity, we neglect the spin-polarization effects for X = Cr, Mn and Fe, being discussed in Refs. 33) and 34). We concentrate to investigate the general features of the long-ranged LLD effects around a single impurity X in Al.

3.1 Lattice parameters of X (= H∼Sn) and Al₃X metals, and HF-forces at the 1st-nn atom around a single impurity X in Al

Figure 2 shows the relation between the equilibrium lattice parameters of X (= H∼Sn) and Al₃X metals in fcc structure, and the HF-forces at the 1st-nn atom (1st-nn HF force) at the ideal atomic position around a single impurity X in Al, all of which have been obtained by the simple NSGGA-FPKKR calculations. The accuracy of the simple NSGGA calculations will be discussed in Sect. 3.2. The equilibrium lattice parameters (= $2\sqrt{2} $ × atomic radii) estimated from the metallic atomic radii listed in Ref. 24), being determined experimentally or theoretically, are also shown for a comparison. The positive value of the 1st-nn HF-force means repulsion (expansion from the Al equilibrium lattice), while the negative value attraction (compression from the Al equilibrium lattice).

Fig. 2

Calculated results for the equilibrium lattice parameters for X (= H∼Sn) and Al₃X metals in fcc structure and the HF-forces on the 1st-nn Al around a single impurity X in Al. The equilibrium lattice parameters estimated from the metallic atomic radii in Ref. 24) are also shown for a comparison.

It is easily recognized from Fig. 2 that the chemical trend of the lattice parameters of X and Al₃X metals in fcc structure can be understood systematically by the 1st-nn HF-force; the variation in the lattice parameters of X and Al₃X metals in fcc structure, such as an expansion or a compression from the equilibrium lattice parameter of Al metal, is well correlated with the sign and magnitude of the 1st-nn HF-force, which becomes larger with the atomic-size misfit (corresponding to the difference between the lattice parameters of Al and X metals). For example, the largest value of the equilibrium lattice parameter of X = Rb metal agrees very well with the largest positive value of the 1st-nn HF force for X = Rb in Al, while the smallest value of the equilibrium lattice parameter of X = Fe with the largest negative value of the 1st-nn HF force for X = Fe in Al.

The strong repulsion of X = Ne, Ar, and Kr (inert gas elements) and Na, K, and Rb (next to the inert gas elements in the periodic table) in Al is mainly due to the repulsion of the core electrons of the large-radius atoms. According to the present calculations, the repulsion of X = Na, K, and Rb is larger than that of X = Ne, Ar, and Kr, respectively. These results may be understood by considering that the X (= Na, K, and Rb) atoms may become monovalent ions in X metals because the s valence states of X atoms are very delocalized, resulting in the repulsion between the monovalent ions. It is noted that the equilibrium lattice parameter of X metal increases with the radius of the s valence electron. On the other hand, we found the attraction for X = d series in the periodic table; the strongest around X = Fe and Ru (the middle of d series in the periodic table). As discussed in Ref. 22), the strong attraction may be not simply due to the atomic radius, but due to the strong sp-d interaction of Al-X because the atomic-size misfit between Al and X (= d elements) is not so large, as seen in Fig. 2. Thus, we may conclude that the sp-d interaction of X (= d elements) impurities with Al host atoms is very important for the inward relaxation (compression) around X.

3.2 FPKKR calculations for LLD energies in the vicinity of a single impurity X (= Mg, Sc, Cu, Zn, Rb, Zr, and Ru) in Al

In a first approximation for the atomic displacements around a single impurity X in Al, we neglect the distortion of distant neighbors and consider only the relaxation of the 1st-nn atoms around the single impurity X in Al, fixing all other atoms at their ideal positions. For symmetry reasons (O_h symmetry) of the impurity system the twelve 1st-nn atoms relax radially by the same size, i.e., in the ⟨110⟩ directions. It is noted that the semicore states of the 3p states of X = Sc, the 3d states of X = Zn, the 4p states of X = Zr and Ru, and the 4sp states of X = Rb, are treated as valence states in the present calculations because the semicore states become important for the atomic displacements, as discussed in Ref. 10).

Figure 3 shows the total-energy variation and the HF radial force at the 1st-nn atom as a function of the displacement of the 1st-nn atom for a single impurity X in Al. The calculated results based on the LSDA, GGA and NSGGA are shown in Fig. 3, in order to examine the differences among them. The LLD energy is defined by the total-energy difference between the ideal and equilibrium atomic positions. It is noted that the total-energy variation shows a parabolic behavior and the HF-force changes linearly as a function of the change of the 1st-nn distance, for all the X impurities, and that the magnitude of the change of the 1st-nn distance is less than 2% of the 1st-nn distance, except for X = Rb (∼6%) and Ru (∼3%). These results suggest that the Kanzaki model for the long-ranged LLD energy for X (= Mg, Sc, Cu, Zn, and Zr) in Al, based on the harmonic approximation for the atomic displacements, may be an accurate approximation for the direct FPKKR calculations, as shown in Sect. 3.3.

Fig. 3

Total-energy variation and HF-force on the 1st-nn host atom as a function of the displacement of the 1st-nn host atom, for X = Mg (a), Cu (b), Zn (c), Sc (d), Zr (e), Ru (f), and Rb (g) in Al. The FPKKR calculation results based on the LSDA, GGA and NSGGA are shown by blue circles/lines, red circles/lines, and black crosses/dotted lines, respectively. The numerical scale for the HF forces is the same as that for the total-energy variation (the left-hand side). The lattice parameter of Al is fixed at 7.70 a_B.

We now discuss the details of the FPKKR calculation results. First we found that, in the cases of the self-consistent LSDA and GGA calculations, the atomic positions at the minimum of the total energies agree very well (within the error of ∼0.1% of the interatomic distance) with those obtained by the zero-force condition, as seen in Figs. 3 and listed in Table 1(a). Second we found that the GC effects over the LSDA are very weak for X = Mg (sp elements with delocalized 3sp states), slightly strong for X = Sc and Zr (early d elements with the delocalized d states and the 3p or 4p semicore states), and fairy strong for X = Cu (with strongly localized d states) and X = Zn (with the 3d semicore states similar to the 3d states of Cu), and Ru (at the middle of 4d series and with the strong sp-d interaction due to the appropriately localized d states and the 4p semicore states). These results may be easily understood because the GC effects become stronger with the interaction of the localized d states (and also semicore states) of a single impurity X with the 3sp states of the neighboring Al atoms. We also found that the GC effects become fairy strong for X = Rb (5sp element). As seen Fig. 3(g), the difference between the LLD energies obtained by LSDA and GGA calculations is as large as 0.003 Ry, although the difference between the atomic displacements is as small as 0.08% of the 1st-nn distance. It is noted the interatomic distance (∼7.7/$\sqrt{2 } $ a_B) between X = Rb atom and its 1st-nn Al atom becomes very short, compared with that (∼12.9/$\sqrt{2 } $ a_B) in Rb metal, resulting in the fairy strong interaction of the 4sp semicore states of X = Rb with the 3sp states of the neighboring Al atoms.

Table 1 Calculated results for (a) the change of the 1st-nn distance (in %) and (b) the LLD energies (in eV), due to only the relaxation of the 1st-nn atoms around a single impurity X (= Mg, Cu, Zn, Sc, Zr, Ru, and Rb) in Al, obtained by the GGA-FPKKR calculations and the Kanzaki model combined with the GGA-FPKKR method. For the change of the 1st-nn distance, obtained by the GGA-FPKKR calculation, we list both the values determined from the zero-force condition and from the minimum of the total energy (in parentheses). For the LLD energies obtained by the GGA-FPKKR calculations, we list the values determined from the minimum of the total energy. The differences between the LLD energies obtained by both the calculations are also listed in parentheses.

Now we discuss the accuracy of the NSGGA treatment for the self-consistent GGA calculations. First, for the total-energy variation, we found that the simple NSGGA calculations reproduce very accurately the LLD energies obtained by the self-consistent GGA calculations, although the discrepancy between them increases with the magnitude of the LLD energy. For example, the discrepancy is less than 0.1 mRy for X = Mg, Sc, Cu, Zn, and Zr with the magnitude of the LLD energy less than 5 mRy. The discrepancy due to the NSGGA is still as small as ∼0.2 mRy and ∼0.3 mRy, even for X = Ru and Rb with the large LLD energies −19 and −66 mRy. These results may be easily understood because the error in the total energy is the second order of the electron-density error if the NSGGA total energy is evaluated by the trial electron-density close to the exact electron-density.²⁹⁾

Second we discuss the accuracy of the NSGGA treatment for the HF-force calculations. We also found in Fig. 3 that, for X = Mg, Sc, and Zr with the delocalized valence states (3sp states of Mg and d states of Sc and Zr), the equilibrium atomic positions obtained by the zero-force condition of the NSGGA calculations agree fairy well (the error of the interatomic distance = 0.11, 0.12, and 0.14%) with those obtained by the total-energy minimum of the GGA calculations. On the other hand, the disagreement between them becomes fairy larger with the GC effects of a single impurity X in Al. For X = Cu (with strongly localized 3d states), Zn (with 3d semicore states), and Rb (with 4sp semicore states), the disagreement becomes as large as 0.32, 0.32, and 0.31%, respectively. For X = Ru, being located at the middle of 4d series in the periodic table, the disagreement is 0.21%, which is almost the middle value between those of the early (for example, Sc) and late (for example, Cu) 3d impurities. These results may also be understood easily because the GC effects increase with the localization of the valence electron-density of a single impurity X (and also with the interaction between the semicore states of a single impurity X and the 3sp states of the neighboring Al atoms, as discussed above) and the error in the NSGGA calculations for the HF-force is the first-order of the electron-density error.

3.3 Accuracy of Kanzaki model combined with GGA-FPKKR method: Comparison with direct GGA-FPKKR calculation results

In order to examine the accuracy of the present Kanzaki model as an approximation for the direct GGA-FPKKR calculations, we now compare the LLD energies obtained by the present Kanzaki model calculations with those by the direct GGA-FPKKR calculations, shown in Figs. 3. It is noted that only the relaxation of the 1st-nn atoms around a single X (Mg, Cu, Zn, Sc, Zr, Ru, and Rb) impurity is considered in both the calculations. We first compare the atomic displacements (= change of the 1st-nn distance) obtained by both the calculations, listed in Table 1(a). The agreement between both the calculated results are very nice, although the discrepancy becomes larger with the atomic displacement. For example, the discrepancy is as small as 0.01% for X-Zn with the small atomic displacement (−0.20%), while as large as 0.40% for X = Rb with the large atomic displacement (5.64%). We second compare the calculated LLD energies by both the calculations, listed in Table 1(b). The discrepancies between the calculated values, due to the harmonic approximation for the atomic displacements, are also listed in Table 1(b). We find the nice agreement between both the calculated results, although the discrepancy also becomes larger with the atomic displacement. For example, the discrepancy is as small as 0.14 meV for X = Zn because of the small atomic displacement (the change of the interatomic distance = −0.19%). On the other hand, the discrepancy is as large as 0.014 eV for X = Rb because of the large atomic displacement (the change of the interatomic distance = 6.04%). It is also noted that the discrepancy is less than ∼0.001 eV for X = Mg, Cu, Zn, Sc, and Zr, for which the change of the interatomic distance is distributed in −1.53∼1.16%, and as slightly large as 0.005 eV for X = Ru with the strong sp-d interaction (the change of the interatomic distance = −3.20%).

3.4 Long-ranged characters of HF-forces, atomic displacements, and LLD energies around a single impurity X (= H∼Sn) in Al, obtained by Kanzaki model combined with GGA-FPKKR method

In order to investigate the long-ranged characters of the LLD effects due to a single impurity X (= Mg, Cu, Zn, Sc, Zr, Ru, and Rb) in Al, we calculate the atomic displacements and the LLD energies over ∼10000 Al atoms around a single impurity X in Al. As explained in Sect. 2.2, we change n (= the size of the cubic cell) from 1 to 13. As a result, the total number (N) of the displaced Al atoms in the cubic cell changes from 12 to 9840, corresponding to n = 1∼13.

First we discuss the distance dependence of the calculated HF-forces. Figure 4 shows the distance dependence of the HF forces up to the 10th-nn Al atoms on the ideal positions around a single impurity (a) X = Mg, Cu and Zn, (b) X = Sc, Rb, Zr and Ru. It is shown that the HF-force is dominant at the 1st-nn atom and decreases rapidly with the interatomic distance. From the present calculation results, the HF-force may be expected to be very weak at the atomic positions beyond the 10th-nn sites. It is also shown that the decreasing behavior of the distance dependence of the HF-force (Fig. 4(b)) is oscillating for X = Sc, Zr, and Ru (d elements), differently from the monotonical decrease for X = Mg and Zn (Fig. 4(a)). For X = Cu (Fig. 4(a)) with the strongly localized d states, we may see the very weak oscillating behavior. On the other hand, for X = Ru (Fig. 4(b)), we can see clearly the oscillating behavior because of the strong sp-d interaction of the appropriately localized d states of X = Ru with the 3sp states of neighboring Al atoms.^22,34) For the early d elements of X = Sc and Zr (Fig. 4(b)), the oscillating behavior is slightly weak because of the weak sp-d interaction. The 3d and 4d orbitals of Sc and Zr are delocalized and the local density of states (LDOS) at the Fermi level are low with the band-broadening of d states.²²⁾ As a result, the sp-d interaction of the d states of Sc and Zr impurities with the 3sp states of the neighboring Al atoms may be weaker than that of Al–Ru, but stronger than that of Al–Cu, and the direct (repulsive) interaction of Sc or Zr with the neighboring Al atoms remain up to the 2nd-nn, as have been discussed in Ref. 22). It is noted that the HF-force at the 2nd-nn site remains relatively strong repulsion for X = Sc, Zr, as seen in Fig. 4(a). For X = Rb (Fig. 4(b)), we found the strong repulsion at the 1st-nn atom and the slightly strong attractions at the 2nd-nn and 4th-nn sites. The main cause of the strong repulsion at the 1st-nn atom is obviously mainly due to the strong repulsion from the core-electrons of Rb, as discussed in Sect. 3.1.

Fig. 4

Distance dependence of the HF-forces on the ideal atomic positions up to the 10th-nn host atoms around a single impurity X in Al, obtained by the GGA-FPKKR calculations: (a) X = Mg, Cu, and Zn, and (b) X = Sc, Rb, Zr, and Ru. There are two equivalent sites for the 9th-nn host atoms. The lattice parameter of Al is fixed to be a = 7.70 a_B.

Second we discuss the calculated results for the LLD energies of a single impurity X in Al. The N(n)-dependence of the LLD energies are listed in Table 2(a). It is noted that the LLD effects due to the atomic relaxations at the distant neighbors beyond the 1st-nn atoms around a single impurity X in Al become stronger with the atomic displacement of the 1st-nn Al atom (listed in Table 1(a)). For example, the LLD energy due to the relaxation at the distant neighbors beyond the 1st-nn sites is as large as −0.22 eV for X = Rb with the largest atomic displacement of the 1st-nn Al atom, while it is as small as −0.4 meV for X = Zn with the small atomic displacement of the 1st-nn Al atom. For X = Mg and Cu with the relatively large atomic displacement, the LLD energy due to the relaxations at the distant neighbors beyond the 1st-nn sites is as large as −12 meV and −15 meV. The value of −12 meV for X = Mg is as large as 35% of the LLD energy (−34 meV) obtained by only the relaxion at the 1st-nn atoms around X = Mg in Al. For X = Ru with the strong sp-d interaction, we found that the LLD energy due to the relaxations at the distant neighbors beyond the 1st-nn sites becomes as large as −67 meV. For X = Sc and Zr, we also found that the LLD energy due to the relaxations at the distant neighbors beyond the 1st-nn sites becomes as large as −20 meV and −29 meV, which may be larger than those of X = Mg and Zn, because of the long-ranged HF-forces due to the sp-d interaction, although the sp-d interaction of Al-X (X = Sc and Zr) is not so strong as that of Al–Ru.

Table 2 Long-ranged character of (a) LLD energies (in eV) and (b) atomic displacements of the 1st-nn host atoms (in %) around a single impurity X (= Mg, Cu, Zn, Sc, Zr, Ru, and Rb) in Al, obtained by the Kanzaki model combined with the GGA-FPKKR method. The LLD energies and the atomic displacements are listed as the dependence on N (n), where N is the total number of the displaced atoms (N = 12∼9840) in the cubic cell, and n (= 1∼13) indicates the size of the cubic cell (the length of one side = na), all the Al atoms in which are allowed to relax.

Except for X = Ru and Rb with the large atomic displacement, the LLD energies are almost converged within 0.01 meV by taking into account up to the atomic displacements in the cubic cell of n = 13, corresponding to the total number of relaxed atoms = 9840. For X = Rb and Ru, we can also get the convergence within 0.04 meV by taking into account the atomic displacements in the cubic cell of n = 13 (N = 9840).

Third we discuss the fundamental features of the distance dependence of the atomic displacements around a single impurity X in Al, shown in Figs. 5. We find that the distance dependence of the atomic displacements is correlated very well with that of the HF-forces; the decreasing behavior of the atomic displacements is monotonical with the interatomic distance, for X = Mg, Cu, and Zn, while oscillatory for X = Sc, Zr, and Ru. For X = Rb, we also find the oscillating behavior of the atomic displacements around X = Rb because there exists the strong repulsion at the 1st-nn atom and the slightly strong attractions at the 2nd- and 4th-nn sites.

Fig. 5

Atomic displacements of the host atoms in the neighborhood of a single impurity X in Al, up to the 20th-nn, obtained by the Kanzaki model combined with the GGA-FPKKR method: (a) X = Mg, (b) X = Cu, (c) X = Zn, (d) X = Sc, (e) X = Zr, (f) X = Ru, and (g) X = Rb. There are two equivalent sites for the 9th-, 13th-, 16th-, 17th-, and 18th-nn host atoms. The HF-forces up to the 10th-nn host atoms (in Figs. 4) are also shown in order to compare with the atomic displacements. The numerical scale for the HF forces is the same as that for the atomic displacements (the left-hand side).

We also list the N(n)-dependence of the atomic displacements of the 1st-nn Al atoms around X in Table 2(b). It is found that the N(n)-dependence of the atomic displacements increases with the atomic-size misfit. For example, the change of the 1st-nn distance, due to the atomic relaxations at the distant neighbors beyond the 1st-nn sites, is as small as 0.06% (of the interatomic distance) for X = Zn (with the small atomic-size misfit), while as large as 1.31% for X = Rb (with the large atomic-size misfit). By using the cubic-cell of n = 7 (N = 1684), we can get the convergence less than ∼0.01%, even for X = Rb.

4. Summary and Future Problem

In order to calculate the long-ranged LLD energy of X_n in the Al-rich AlX disordered alloys, as the deviation from the global volume relaxations of the Al-rich AlX alloy, we proposed the Kanzaki model combined with the GGA-FPKKR method, as an approximation of the direct GGA-FPKKR calculations, which is based on the harmonic approximation for the atomic displacements and enables us to calculate the long-ranged LLD energies (over ∼10000 atoms) in the disordered alloys.

We first showed the accuracy of the Kanzaki model combined with the GGA-FPKKR method, which reproduces very accurately the LLD energies obtained by the direct FPKKR calculations, for a single impurity X in Al, such as X = Mg, Cu, Zn, Sc, and Zr with the atomic displacement less than 2% of the interatomic distance. We second clarified the fundamental features of the long-ranged LLD energies (over ∼10000 host atoms) around a single impurity X (= Mg, Cu, Zn, Sc, Rb, Zr, and Ru) in Al, such as a distance dependence from a single impurity X in Al, by using the present Kanzaki model. It is noted that the LLD energies around a single impurity X in Al correspond to the LLD part in the 1-body IEs in the RSCE from a dilute limit, for the internal energies of the Al-rich AlX disordered alloys.^16–19) We showed that the LLD energy in the 1-body IE becomes negatively larger and more long-ranged with the atomic-size misfit between Al and X elements (especially for X = Rb) and also with the sp-d interaction of Al-X (especially for X = Ru at the middle of the 4d series of the periodic Table). We also found that the distance dependence of the HF-forces at the atoms around a single impurity X (= d element) in Al shows the oscillating behavior and that the distance dependence of the atomic displacement is correlated very well with that of the HF-forces.

We now start to calculate the LLD part in the n-body IE of X_n (X = H∼Sn, n = 2∼4) impurities in the Al-rich AlX alloys, by using the RSCE from a dilute limit.^16–18) It is noted that the LLD part in the n-body IE of X_n, including the anisotropy of the atomic displacements, can be calculated uniquely and successively from 1-body to 4-body by the combination of the long-ranged LLD energies of X_n (n = 1∼4) impurities in Al. We will expect that the LLD part in the n-body IE of X_n in Al becomes negatively larger and more long-ranged with the atomic-size misfit between Al and X, and may become smaller with n, as have been shown for the CI part in the n-body IE of X_n in Al.²²⁾

Before closing the present paper, we want to discuss the importance of the long-ranged LLD energies, for example, in the age-hardenable Al-rich AlMg alloys. The CI part in the effective 1st-nn 2-body IE (including up to the 10th-nn),²²⁾ is repulsive (∼0.04 eV), so that a simple consideration of the CI contribution alone contradicts completely the experimental results (segregation at low temperatures). We found that the effective 2-body IE becomes attractive (∼−0.01 eV) by adding the effective 2-body LLD energy (∼−0.05 eV), including up to the 4th-nn, which were obtained by the direct GGA-FPKKR calculations with the restriction on the displacement of only the 1st-nn atoms around Mg₂ impurities in Al.^16,17) According to the preliminary free energy calculations with the CVMT and CVMTO for the configurational entropy,^16,17) in order to reproduce the observed solvus temperatures (400∼600 K) of Al-rich AlMg alloys,³⁵⁾ we may need the additional attraction (∼−0.02 eV) for the effective 2-body IE. We may expect that most of this additional attraction can be obtained by the long-ranged LLD effects at the distant neighbors beyond the 1st-nn sites, because the additional attraction of a single impurity X = Mg in Al becomes as large as −0.012 eV (= 35% of the LLD energy (−0.034 eV) for only the relaxation of the 1st-nn atoms), as listed in Table 2(a). We may also expect the attractive effects for the LLD part in the n-body IE (n ≥ 3). Thus, the Kanzaki model calculations for the long-ranged LLD energies in the Al-rich AlMg alloys may be very promising to obtain the additional attraction for the effective 2-body IE and also the LLD part in the n-body IE (n ≥ 3), being neglected in the previous approach,^16,17) and reproduce the observed solvus temperatures of the Al-rich AlX (X = Mg and Cu) alloys.

Acknowledgement

The authors are grateful for the financial support from the Ministry of Education, Culture, Sports, Science and Technology (JSPS KAKENHI Grand Nos. 23560792 and 24560901).

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