Characteristic atom occupation patterns of Au₃Cu, AuCu₃, AuCuI and AuCuII based on experimental data of disordered alloys

XIE You-qing^{1, 2, 3}, LI Yan-fen^{1, 2, 3}, LIU Xin-bi^{1, 2, 3}, LI Xiao-bo⁴, PENG Hong-jian^{1, 2, 3}, NIE Yao-zhuang^{1, 2, 3}

1. School of Materials Science and Engineering, Central South University, Changsha 410083, China;

2. Powder Metallurgy Research Institute, Central South University, Changsha 410083, China;

3. State Key Laboratory for Powder Metallurgy, Central South University, Changsha 410083, China;

4. College of Mechanical Engineering, Xiangtan University, Xiangtan 411105, China

Received 14 April 2010; accepted 10 March 2011

Abstract: The potential energies, volumes and electronic structures of characteristic atoms coordinated by neighboring configurations were obtained from the experimental heats of formation and lattice parameters of disordered Au_1-xCu_x alloys. From characteristic atom occupation (CAO) patterns of L1₂-Au₃Cu, L1₂-AuCu₃ and L1₀-AuCu compounds, their electronic structures, volumetric and energetic properties were calculated. The CAO pattern of Johasson-Linde(J-L) model shows that the transition AuCuI→AuCuII is an exothermic and volume contraction reaction, which is opposite from experimental phenomena. According to CAO pattern of Guymont-Feutelais-Legendre(G-F-L) model, the AuCuII cell consists of two periodic antidirection (PAD) AuCuI regions and two PAD boundary regions. The equations derived from CAO pattern of G-F-L model can be used to calculate energetic properties, volumetric properties and ordering degrees of the PAD AuCuI region and PAD boundary region, as well as corresponding average properties of the AuCuII phase. The results are consistent with experimental phenomena.

Key words: Au-Cu system; intermetallics; electronic structure; crystalline structure; characteristic atom occupation pattern

1 Introduction

An alloy system contains three structure levels: phase level of organizations, atomic level of phases and electronic level of atoms. In order to get an entirely understanding of the alloy systems, to establish phase diagrams and to search a method for designing of alloys, the systematic science of alloys (SSA) has been established[1-5].

One of the main philosophic viewpoints in the SSA framework is that “A diversity of structures, properties and features of matter or non-matter systems should be attributed to combination and arrangement of structural units in the structural unit sequence”. For example, the diversity of atoms in the atomic system is attributed to arrangement of electrons in the electronic orbital sequence; the diversity of substances in the matter system is attributed to composition of elements in the periodic sequence of elements; the diversity of geometric figures in the geometry is attributed to connection of straight lines in the straight line sequence with various lengths and curved lines in round sequence with various curvature radii; the diversity of species in the biological system is attributed to splice of the genes in the gene sequence; the diversity of music compositions in the music system is attributed to combination and arrangement of notes in the music note sequence.

In the fcc-based lattice Au-Cu system, each atom is coordinated by a nearest configuration [(I-i)Au, iCu], where i denotes the number of Cu atoms and I is coordination number and equal to 12. In the L1₀-AuCu compound, each Au atom is coordinated by configuration [4Au, 8Cu], and named as  characteristic atom; each Cu atom is coordinated by configuration [8Au, 4Cu], and named as characteristic atom. Each characteristic atom has its own characters: potential energy, volume and electronic structure.

The crystalline structure of L1₀-AuCu compound can be described by combination of two superlattices, of which one is occupied by  atoms, another is occupied by  atoms. When L1₀-AuCu disorders to form the random Au_0.5Cu_0.5 alloy, the  and  atoms are split, respectively, into the  and  characteristic atom sequences, of which the concentrations  and  can be calculated[3]. According to this basic analysis, three interconnected models have been provided for constructing diversity of structures and properties of alloy phases[5-8]:

1) The basic cluster overlapping (BCO) model. The structural units are a pair of basic cluster sequences, the first and second neighbor configurations and each cluster consists of a central atom.

2) The characteristic atom arranging (CAA) model. The structural units are a pair of characteristic atom sequences, and each characteristic atom is the central atom of a specific basic cluster.

3) The characteristic crystal mixing (CCM) model. The structure units are a pair of factitious characteristic crystal sequences, and each characteristic crystal consists of the same characteristic atoms with the identical potential energy, identical volume and identical electronic structure.

These three interconnected models were proposed in order to overcome disappointments of the atomic pair interaction model and central atom model[9], four-sublattice model and effective cluster interaction model[10-14].

One of the main methods in the SSA framework is that “the whole can be reproduced from a few parts” So the whole of a tree can be reproduced respectively from a seed, a leaf or a branch of the tree, and the whole of a sheep can be reproduced respectively from an egg cell or a body cell of the sheep in biologic systems; the whole information of an alloy system can be reproduced from information of a few disordered alloys or a few intermetallics, which may be obtained by experimental measurements or the first-principle calculations. This means that “the total potential energies and total volumes of a few alloys can be separated into the potential energy sequence and volume sequence of the characteristic atoms, from which the whole information about details of energetic and volumetric properties, electronic and crystalline structures of all alloy phases, as well as phase diagram of the alloy system can be reproduced”.

The Au-Cu system has an extensive history in modern time on the metallic materials science. It has been considered a good platform for nearly all theories of alloys and nearly all experimental techniques to study electronic and crystalline structures of phases, order-disorder phase transformation and phase diagram. The SSA framework differs from previous theories mainly on two aspects: 1) it is established explicitly based on the philosophic viewpoints and methods; 2) the potential energies and volumes of characteristic atom sequences can be separated out from the total energy and total volume of alloy phase. The previous theories are the framework of total energy and total volume, which cannot be separated. Now, it is also used as the platform systematically to present the SSA framework through a series of papers. In the next section the main equations concerning with this paper are presented. According to the potential energies and volumes of characteristic atom sequences, the electronic structures, lattice constants, cohesive energies, potential energy curves and Debye temperatures of characteristic crystal sequences have been obtained by valence bond(VB) theory. According to the CAO patterns of the crystalline structures of observed L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds, the obtained electronic structures and details of energetic and volumetric properties of these compounds are presented. According to the CAO patterns of the crystalline structures described by two models of the equiatomic AuCuII, the calculated volumetric properties, energetic properties and ordering degrees of AuCuII phase and its periodic antidirection(PAD) boundary region are shown.

2 Main equations

2.1 ε- and v-functions of characteristic atom sequences

Only considering the influence of the nearest neighboring configuration [(I-i)Au, iCu] on the central characteristic atoms, the potential energy ε-function and volume V-function of characteristic atom sequences are obtained[15-17]:

             (1)

            (2)

When considering influence of the first and second neighboring configuration {[(I-i)Au, iCu], [(J-j)Au, jCu]}, the ε- and V-functions can be modified as follows:

   (3)

        (4)

where, J=6; j is the number of Cu atoms in the second neighboring configuration. , , , , , , , .

2.2 Description for electronic structures of characteristic crystals

In the valence bond (VB) theory, the electronic structure of a characteristic crystal, i.e. characteristic atom, is illustrated by valence electrons of one-atom state  hybridized by several basic atom states in the basic atom state  sequence:

                                 (5)

The hybridized composition c_k can be found from combination of three basic states (see section 3.1.2).

If ， and  denote respectively the numbers of covalent electrons of s, p and d orbitals in the k basic atom state; ,  and  denote respectively the number of non-valence d electrons, free s and p electrons, from which the OEO numbers of the characteristic crystal can be obtained by following expressions:

   (6)

where R_k, which is the single bond radius, can be obtained from Pauling’s equation[18] slightly modified[19-20]:

                    (7)

2.3 Bond length and lattice constant equations

Pauling’s bond length equation should be suitable to various bonds in the characteristic crystal, we have (in nm):

                            (8)

where s=1, 2 and 3; r₁, r₂, r₃ and I₁, I₂, I₃ denote respectively the length values and the numbers of the first, second and third neighboring bonds; n₁, n₂ and n₃ denote respectively the numbers of pairs of covalent electrons on the relative bonds; β is Pauling’s coefficient[18].

For a fcc crystal, the lattice constant equation can be derived from Eq.(8)[21-22]:

        (9)

where s=1, 2, 3; , G₂=1, ; n_c is obtained by Eq.(6). It is a convergent transcendental function of the lattice constant.

2.4 Cohesive energy equation and MAI potential function

Considering contributions of the covalent and near free electrons to cohesive energy, the cohesive energy equation (E_c) and potential function (w(r)) with many-atom interactions (MAI) of characteristic crystals have been established[23] , which are (in kJ/mol):

             (10)

          (11)

where

      (12)

r₀ and r may denote any kind of bond length, but in the present work, they express the shortest lengths of the crystal in equilibrium and nonequilibrium, respectively; f and f ′ are respectively the bonding capacities of the hybrid covalent electrons and near free electrons. The constants n′ and g should be taken respectively as 4 and 1 for Au and Cu elements.

The exponent x in the MAI potential function is obtained by Debye theory[22-23].

                (13)

where θ is Debye temperature; k_B is the Boltzman’s constant;  is the Planck’s constant; m is atomic mass; j and n are respectively the multiple of the half cutoff wave length and another exponent of the MAI potential function, which can be obtained from two experimental values of the linear thermal expansion coefficients at two temperatures.

The Debye temperature  of the  characteristic crystal is approximately obtained by following equation.

           (14)

where ,and , as well as  and are respectively potential energy, volume and Debye temperature of the pure metal and -characteristic crystal of α-component at 0 K.

2.5 Additive law of characteristic atoms and volume mismatch

The intermetallic compounds can be thought to be formed by arranging characteristic atoms. When considering the influence of the first and second neighbor configurations {[(I-i)Au, iCu]，[(J-j)Au, jCu]}, the total state , total potential energy E and total volume V of a cell for compounds can be obtained by the additive law of characteristic atoms (simplified as CAA-law):

                 (15)

The average atomic volume mismatch degree between characteristic atoms inside α component is

     (16)

The average atomic volume mismatch degree between components Au and Cu is

     (17)

The total average atomic volume mismatch degree of a compound is

                 (18)

In Eqs.(15)-(18), , ,  and  are, respectively, electronic structure, potential energy, volume and number of the  characteristic atom in a cell;  is the average atomic volume of the α-component; n_α and n_T are respectively the number of atoms of the α-component and total number of atoms in a cell; v_a is the average atomic volume of the compound.

In a similar way to obtain various volume mismatch degrees, we can obtain corresponding potential energy mismatch degree: ,,.

3 Results

3.1 Structures and properties of characteristic crystals (atoms)

3.1.1 Basic atomic states and corresponding fcc-crystals

According to covalent electron number (d_c, s_c), free electron number (s_f, p_f) and non-valence electron number (d_n) in outer shell of a basic state atom, the single bond radius (R), lattice constant (a) and cohesive energy (E_c) of fcc-crystals formed respectively by the same Au and the same Cu basic state () atoms can be calculated by Eqs.(6)-(13). Table 1 shows these results of some basic states.

3.1.2 Electronic structures of characteristic crystals

The precise solution (hybridized composition) of electronic structure of each characteristic crystal can be found from combination of three basic states, through the cross point of the equilattice constant a-line and equicohesive energy E_c-line of this characteristic crystal. The solutions of electronic structures of primary  and  characteristic crystals are shown in Fig.1.

The electronic structures of pure Au and Cu metals, i.e. primary characteristic crystals, were also calculated from linear muffintin-tin orbitals(LMTO) method in the energy band (EB) theory by us and were calculated from linear rigorous cell (LRC) method by ECKARDT et al[24]. In their opinion: “as a result of the strong s-p hybridization the occupation of the formerly unoccupied p level becomes practically equal to that of the s level”. These results are approaching to ones by VB theory:

Table 1 Some Au and Cu basic atomic states in Au and Cu basic atom state sequences and corresponding fcc-crystals

Fig.1 Hybridized compositions of basic states and electronic structures for primary  and  characteristic crystals at 0 K: (a)(b)([Xe] is Xenon electronic structure; [Ar] is Argon electronic structure)

The lattice constants and cohesive energies calculated from LMTO method are respectively 0.403 9 nm and 438 kJ/mol for fcc pure Au metal and 0.355 1 nm and 419.19 kJ/mol for fcc pure Cu metal. These results are not in good agreement with experiments and not satisfied to calculate thermodynamic properties of alloys. According to electronic structures of primary  and  characteristic crystals, their potential energy curves are calculated by Eqs.(8)-(11) and shown in Fig.2.

3.1.3 Characteristic crystal sequences

According to the potential energies and volumes of  and  characteristic atoms, the electronic structures, single bond radii, cohesive energies and Debye temperatures of  and  characteristic crystals have been calculated by VB theory and are listed in Table 2. The nature of variations in potential energies and volumes of  and  characteristic atoms can be expounded.

1) For  characteristic crystals, the atomic potential energy lowers with increasing the i number of Cu-atoms in the configuration [(I-i)Au, iCu], essentially due to both increase of total valence electrons  and reduction of atomic volume; the atomic volume reduces, essentially due to both increase of total covalent electrons  and decrease of Pauling’s single bond radius R, which is attributed to increase of (Eq.(7)).

Fig.2 Potential energy curves of primary  (a) and  (b) characteristic crystals at 0 K

Table 2 Valence electrons in outer shell, atomic potential energies (ε), atomic volumes (v), single bond radii (R), cohesive energies (E_c) and Debye temperatures (θ) of  and  characteristic crystal sequences in Au-Cu system at 0 K

2) For  characteristic crystals, the atomic potential energy lowers with increasing the (I-i) number of Au-atoms in the configuration [(I-i)Au, iCu], mainly due to the increase of total valence electrons, even the increase of the atomic volume rises; the atomic volume rises mainly due to the increase of Pauling’s single bond radius R, which is attributed to decrease of , even the increase of total covalence electrons.

According to the potential energies and volumes of  and  characteristic atoms coordinated by configuration {[(I-i)Au, iCu], [(J-j)Au, jCu]}, the electronic structures and properties of  and  characteristic crystals consisting of  and atoms are also calculated, but here these results are not listed.

In Table 1, , . The effect of the second neighboring configuration [(J-j)Au, jCu] on these properties can not be considered calculating variations of volumetric and energetic properties of alloys with composition and ordering degree.

3.2 L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds

According to CAO patterns of observed L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds, we can obtain electronic structures, potential energy wave planes, details of volumetric and energetic properties of these compounds (see Fig.3 and Table 3), from which the following information is obtained.

The electronic structures of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds are described by those of characteristic atoms (see Table 2):

Fig.3 CAO patterns and potential energy wave (111) planes of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃: (a) L1₂-Au₃Cu; (b) L1₀-AuCu; (c) L1₂-AuCu₃; (d)(e)(f)

Table 3 Volumetric properties and energetic properties of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds as well as corresponding disordered alloys at 298.15 K

We have obtained not only the average atomic volumes (v) of these three compounds and corresponding disordered alloys, but also the average atomic volumes (v_Au and v_Cu) of Au- and Cu-components in ?³/atom; The average atomic volume of the compound is smaller than that of the corresponding disordered alloy.

We obtained not only the average atomic volumes of formation(Δv^m), but also the average atomic volumes of formation (,) of Au- and Cu-components in ?³/atom. The average atomic volume of the compound is smaller than that of the corresponding disordered alloy, to which the contribution of the Au-component is negative, the contribution of the Cu-component is positive, and Δv^m>0.

We have obtained not only the ordering (or excess) average atomic volumes (Δv^ex), but also the ordering average atomic volumes (,) of Au- and Cu-components in ?³/atom, to which the contribution of the Au-component is negative, the contribution of the Cu-component is positive, and Δv^ex<0.

We obtained not only the total average atomic volume mismatch degree () of these three compounds and corresponding disordered alloys, but also the average atomic volume mismatch degrees (Δm_V) between Au- and Cu-components, as well as their average atomic volume mismatch degrees (,) between characteristic atoms inside the Au- and Cu-components, respectively, in ?³/atom. The total average atomic volume mismatch degree of the compound is larger than that of the corresponding disordered alloy.

In a similar way to volumetric properties, we obtained the energetic properties: average atomic potential energies (ε, in eV/aton), average atomic potential energies of formation (), ordering average atomic potential energies ()  and total average atomic potential energies mismatch degrees () of these three compounds and corresponding disordered alloys, as well as corresponding energetic properties (,;,;,;,) of Au and Cu components.

We obtained the cohesive energies (E_c, in J/mol), heat of formation (ΔH^m, in J/mol), volume of formation (ΔV^m, in ?³/mol) and lattice parameters (a, b, c, in ?) of these three compounds and corresponding disordered alloys.

The interpretations of the symbols and units have been given in this section. The experimental values are listed in parentheses.

3.3 Volumetric and energetic properties of AuCuII

3.3.1 CAO pattern of J-L crystalline model

The crystalline structure of AuCuII compound was firstly investigated through X-ray diffractometry by JOHASSON and LINDE[29]. From their results we can know that the AuCuII is a periodic antiphase commensurate structure, stacking of L1₀ cells along b equal to 10a, of space group Imam, and the number (m) of L1₀ cells between two successive antiphase boundary planes is equal to 5. This structure is thus conveniently described as follows: every five order L1₀ cells along b axis, an antiphase boundary of vector [1/2 0 1/2] operates, so that after a 10a spacing, the cell is restored (see Fig.4(a1)).

According to J-L model, the CAO pattern has been drawn up (see Fig.4(a2)). It shows that the AuCuII cell consists of two PAD AuCuI regions and two PAD boundary regions. Each PAD AuCuI region contains 6 (+) planes perpendicular to b axis, in which the and atoms are arranged alternatively by face center form. Each PAD boundary region contains 4 (+) planes, in each plane, the and  atoms are arranged alternatively by face center form, the volume and potential energy of each AuCuII cell can be calculated by following equations:

               (19)

              (20)

The calculated volumetric and energetic properties of PAD AuCuI region are listed in Table 3, and the calculated volumetric and energetic properties of AuCuII and PAD boundary region are listed in Table 4. From Tables 3 and 4, we can obtain following knowledge.

When the AuCuI→AuCuII transformation occurs, the and  atoms with higher potential energies (=-3.9210 eV/atom, =-3.5635 eV/aton) change respectively into the and  atoms with lower potential energies (=-3.9255 eV/atom, =-3.5669 eV/atom) in the PAD boundary region, the transition enthalpy ΔH(AuCuI→AuCuII) is negative (-152.7 J/mol), namely, it is an exothermic reaction, that is opposite from experimental phenomenon.

The =16.3181 ?³/atom, which is smaller than =16.3437 ?/atom, and the =12.6945 ?³/atom, which is larger than =12.6855 ?³/atom, but the average atomic volume (14.5113 ?³) of AuCuII is smaller than the average atomic volume (14.5146 ?³) of AuCuI, then, the transition volume Δv(AuCuI→AuCuII) is negative (-0.0033 ?³/atom), that is opposite from the experimental phenomenon too. Therefore, the J-L model can not be used to describe the AuCuII compound.

3.3.2 CAO pattern of G-F-L crystalline model

The in-situ X-ray diffraction experiments and in-situ temperature observations in transmission electron microscopy on stoichiometric AuCu alloy were performed[28-30]. From their results we can know the follows.

Fig.4 Crystalline structure of J-L model for AuCuII (a1), CAO pattern of J-L model (a2), crystalline structure of G-F-L model (b1) and CAO pattern of G-F-L model (b2)

Table 4 Volumetric properties and energetic properties of AuCuII and PAD boundary(B) region described by J-L model, as well as their transition volumes Δv(AuCuI→AuCuII), Δv(AuCuI→PADB) and transition enthalpy ΔH(AuCuI→AuCuII), ΔH(AuCuI→ PADB) (The interpretation of other symbols and units is the same to that of Table 3)

The structure of AuCuII described by G-F-L model is a periodic antiphase incommensurate structure, statistical stacking of L1₀ cells along b axis, which is thus periodic along a and c, but only periodic in the mean along b. At the equiatomic composition, AuCuII seems ordered everywhere except in the neighbourhood of the antiphase boundaries (see Fig.4(b1)).

According to CAO pattern of the G-F-L model of the equiatomic AuCuII (see Fig.4(b2)), the AuCuII cell consists of two PAD AuCuI regions and  two PAD boundary regions. Each AuCuI region, supposing full order (σ=1), contains m (+) planes, at the very least m=6, perpendicular to b axis, in which the and  atoms are arranged alternatively by face center form. Each PAD boundary region contains

planes, at the very least m′=4. In each plane, the average  and average  atoms are arranged alternatively by face center form. But the numbers m=6 and m′=4 are nevertheless still the results of an average. The concentrations  and  of the characteristic atoms  and  are the functions of composition x and ordering degree σ[4], where σ<1. The volume and potential energy of each AuCuII cell described by G-F-L model can be calculated by following equations:

                     (21)

                   (22)

The calculated results are shown in Fig.5 and listed in Table 5. The experimental ΔH and Δv values are given in the corresponding literatures. From Table 5, we can obtain the following knowledge.

1) Even at the stochiometric AuCu, the transformation AuCuI→AuCuII occurs on a range of transition temperatures, which may be mainly attributed to periodic composition fluctuation in micro-regions. It leads to the diffusion of atoms in cooperation with splitting of the energetic and volumetric states of Au- and Cu-characteristic atoms, to form PADB regions with higher potential energy, larger volume and lower ordering degree, and to make compositional gradient match with both side neighboring PAD AuCuI regions. Therefore, it is an energy absorption and volume expansion reaction, which is consistent with the experimental phenomena.

2) According to potential energies of Au and Cu characteristic atoms, which were determined on the basis of experimental heats of formation of disordered Au_1-xCu_x alloys in the compositional ranges 0≤x_Cu≤25% and 75%≤x_Cu≤100% to avoid effect of short-range (SR) ordering degree, the transition enthalpy ΔH of transformation AuCuI→AuCuII as functions of average ordering degree σ of AuCuII and σ(PADB) of the PADB region are calculated. According to experimental transition enthalpies 325.6, 756.1 and 886.5 J/mol, their corresponding ordering degrees of the PADB regions are respectively 0.878, 0.683 and 0.621. If taking m=9 and m′=4, their ordering degrees of the PADB regions decrease, but their average ordering degrees of AuCuII keep consistent.

3) According to the volumes of Au and Cu characteristic atoms obtained from experimental volumes of disordered Au_1-xCu_x alloys, the transition volume Δv of transition AuCuI→AuCuII as functions of average ordering degree σ of AuCuII and σ(PADB) of the PADB region are calculated and also listed in Table 5. According to experimental transition volume of 0.0149 ?/atom, the calculated ordering degrees σ(PADB) is 0.260, which is much smaller than those obtained from experimental transition enthalpies. If the volumes of Au and Cu characteristic atoms are determined on the basis of experimental volumes of L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds, the calculated ordering degree σ(PADB) is 0.569, which is close to 0.593 obtained by experimental transition enthalpy of 886.5 J/mol. This fact tells us that there is effect of SR ordering degree on the volumes of Au and Cu characteristic atoms obtained from experimental volumes of disordered Au_1-xCu_x alloys in the whole composition range.

Fig.5 Calculated ΔH(AuCuI→AuCuII), Δv(AuCuI→AuCuII) as function of ordering degree σ(AuCuII) for transition AuCuI→AuCuII,  as well as ΔH(AuCuI→PADB), Δv(AuCuI→PADB) as function of ordering degree σ(PADB) for transition AuCuI→PADB region at equiatomic AuCu, when AuCuI region has full order, m=6 and m′=4. Here σ denotes σ(AuCuII) for AuCuII or σ(PADB) for PADB region, respectively

Table 5 Transition enthalpy ΔH(AuCuI→AuCuII) and transition volume Δv(AuCuI→AuCuII) and transition average lattice constant Δa(AuCuI→AuCuII) as functions of average ordering degree σ(AuCuII) of AuCuII and ordering degree σ(PADB) of PADB region, when AuCuI region has full order, as m=6, m′=4 and m=9, m′=4

4) In the range of AuCuI→AuCuII transition temperatures, the disorder should spread PAD-AuCuI region with high ordering degree σ(H) and PADB region with low ordering degree σ(L). From Eqs.(21) and (22), it can be known that the property q denoting v and ε can be calculated by the following equation, which is a function of composition x, temperature T and ordering degree σ:

                   (23)

4 Conclusions

1) According to the potential energies and volumes of characteristic atoms separated from the experimental formation heats and lattice constants of disordered Au_1-xCu_x alloys, the electronic structures, potential energy curves lattice constants, cohesive energies and Debye temperatures of corresponding characteristic crystals were obtained, and the nature of variations of their potential energies and volumes with the numbers of Cu-atoms in the neighboring configurations was expounded. The electronic structures of the primary characteristic Au and Cu crystals, i.e., pure fcc Au and Cu metals, obtained by VB theory and EB theory (LRC and LMTO methods) were compared. They are approaching each to each. But the electronic structures of other characteristic crystals (atoms) cannot be obtained by EB theory. Therefore, the VB theory of characteristic crystals has enriched contents of electronic structure theory of alloys.

2) According to CAO patterns of observed L1₂-Au₃Cu, L1₀-AuCu and L1₂-AuCu₃ compounds: the electronic structures, potential energies and volumes of Au and Cu atoms at each lattice point in these compounds; the total electronic structures, total volumes, total potential energies (cohesive energies), total volumes of formation and total enthalpies of formation, total ordering volumes and total ordering enthalpies of these compounds; as well as the corresponding energetic and volumetric properties of Au and Cu components.

3) The CAO pattern of J-L model shows that the transition AuCuI→AuCuII is an exothermic volume contraction reaction, which is opposite from experimental phenomena. The CAO pattern of G-F-L model can be used to explain the structure of AuCuII and to calculate energetic and volumetric properties, and ordering degrees of the PADB region, PAD AuCuI region and AuCuII phase, if there are precise experimental values of enthalpies and volumes of transition AuCuI→AuCuII.

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以无序合金的实验数据为基础的Au₃Cu，AuCu₃，

AuCuI和AuCuII的特征原子占据图

谢佑卿^{1, 2, 3}，李艳芬^{1, 2, 3}，刘心笔^{1, 2, 3}，李晓波⁴，彭红建^{1, 2, 3}，聂耀庄^{1, 2, 3}

1. 中南大学 材料科学与工程学院，长沙 410083；

2. 中南大学 粉末冶金研究院，长沙 410083；

3. 中南大学 粉末冶金国家重点实验室，长沙 410083；

4. 湘潭大学 机械工程学院，湘潭 411105

摘  要：根据无序Au_1-xCu_x合金的实验生成热和晶格常数得到只考虑近邻原子组态的特征原子的势能、体积和电子结构，计算L1₂-Au₃Cu, L1₂-AuCu₃和L1₀-AuCu化合物的特征原子占据(CAO)图、电子结构、能量和体积性质。Johasson-Linde(J-L)模型的CAO图表明，AuCuI→AuCuII转变是一个放热且体积缩小的反应，但正好与实验现象相反。根据Guymont-Feutelais-Legendre(G-F-L)模型的CAO图，AuCuII 晶胞由两个周期反相(PAD)的AuCuI区域和两个PAD边界区域组成；从G-F-L 模型的CAO图得出的公式可用来计算PAD AuCuI区域和PAD 边界区域的能量性质、体积性质和有序度及AuCuII 相的平均性质，计算结果与实验现象相吻合。

关键词：Au-Cu系；金属间化合物；电子结构；晶体结构；特征原子占据图

(Edited by YANG Hua)

Foundation item: Project (50711181) supported by the National Natural Science Foundation of China; Project (2009FJ4016) supported by Natural Science Foundation of Hunan Province, China

Corresponding author: XIE You-qing; Tel: +86-731-88879287; E-mail: xieyouq2000@yahoo.com.cn

DOI: 10.1016/S1003-6326(11)60827-3