ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2011, Vol. 75, No. 9, pp. 1267–1273. © Allerton Press, Inc., 2011. Original Russian Text © Yu.M. Gufan, O.V. Kukin, A.Yu. Smolin, 2011, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2011, Vol. 75, No. 9, pp. 1341–1348.
A New Method for Calculating ThreeParticle Interaction in the Theory of Modulus Elasticity Yu. M. Gufana, O. V. Kukina, and A. Yu. Smolinb a
b
Institute of Physics, Southern Federal University, RostovonDon, 334090 Russia National Research Nuclear University (MEPhI), Volgodonsk Engineering Institute, Volgodonsk, 347340 Russia email:
[email protected]
Abstract—We propose a new method for calculating the potential of multiparticle interaction. Our method considers the energy symmetry for clusters that contain N identical particles with respect to permutation of the number of atoms and free rotation in threedimensional space. As an example, we calculate moduli of thirdorder rigidity for copper considering only the threeparticle interaction. We analyze nine models of energy dependence on the polynomials that form the integral rational basis of invariants (IRBI) for the group G3 = O(3) ⊗ P3. In this work, we use only the simplest relation between energy and the invariants forming the
∑
−6
−12
−n
⎡− A1rik + A2rik + Q j I j ⎤⎦ , where I j is the invariant number j (j = 1, 2, …, 9). IRBI: ε (i, k, l j ) = i,k,l ⎣ The results are in good agreement with the experimental values. The best agreement is observed at n = 2, j = 4: I 4 = ( rik rkl )( rkl rli ) + ( rkl rli )( rli rik ) + ( rli rik )( rik rkl ) .
DOI: 10.3103/S1062873811090103
INTRODUCTION Theoretical simulation of metals’ mechanical properties is an increasingly important tool in search ing for and synthesizing materials with predetermined properties. There are many quantum mechanical models that allow us to calculate the moduli of elastic ity for crystals of the Periodic Table [1–5]. The great variation in measurement results obtained on the basis of these models compelled us to analyze calculation procedures based on the concept of symmetry. It is especially important that the potentials of pair interac tion (firstorder interaction) between atoms is included in the Hamiltonians of all models. These potentials are by definition the result of approximate considerations of the Coulomb and exchange interac 1
tions between the electrons of atoms.
The explicit analytical form of twoparticle inter action potentials is therefore unknown. It is fitted arbi trarily according to semiempirical concepts as to the possible type of interaction between two atoms that 2
form an individual pair [6, 7].
This alone demonstrates the arbitrariness that emerges in such models in calculating the cohesive binding energy, to which (in addition to pair interac tions) the interaction of triplets, quartets, and other multiparticle interactions that cannot be reduced to pair interaction make an additive contribution. As for the effective pair interaction between two identical (pointlike) particles at certain nodes of a crystal lattice, we know a priori only that it is charac terized by axial symmetry and depends on the differ ence between the particles’ coordinates. It is now suf ficient to make a series of precise assertions as to the properties of matter between whose atoms there are only effective pair (central) interactions. First, the energy of pair (central) interactions between atoms depends by definition only on the squares of the inter atomic distances. Second, the cohesive binding energy of the atoms in these crystals is equal to the energy needed for the formation of vacancies, but with the opposite sign. Third, for crystals between whose atoms only pair (central) forces operate, the modulus of sec ondorder rigidity is described by the Cauchy relation [9–11]. For cubic crystals. the Cauchy relations can be written as C12 = C 44.
1 The
energy of exchange interaction for a system consisting of n identical particles is essentially an approximation that considers part of the total Coulomb energy of the system [8]. We can sepa rate this part of the energy into an individual summand since it depends on total spin of the system, but this part of the Coulomb energy is traditionally referred to as independent when discuss ing models. 2 The asymptotic type of pair interaction at great distances has been calculated in a number of works [9].
(1)
The modulus of thirdorder rigidity is described by the Milder relation. The Milder relations for the mod ulus of thirdorder rigidity are
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⎧C112 = C166, ⎨ ⎩C123 = C144 = C 456.
(2)
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The symmetry of pair interaction imposes constraints on modulus of fourthorder rigidity:
⎧C1112 = C1166, ⎪ ⎨C1122 = C1266 = C 4444, ⎪⎩C1123 = C1144 = C1244 = C1456 = C 4466.
(3)
In earlier works, the empirical potentials of three atom interaction (which cannot be reduced to pair interaction) were included in the potential energy in order to eliminate the “nonphysical” Cauchy rela tions. In other words, they used the concept of the full energy of interaction between atoms in the form of the sum of interactions between atoms inside clusters con sisting of different numbers of particles. The first approximation in such a total energy expansion with respect to cluster energies considers only the pair interactions of atoms. The next approximation con siders threeparticle interaction (which cannot be reduced to sums of pair interactions), and so on. Threeparticle interaction is, however, also character ized by additional symmetry that is atypical of crystal lattices: at any disposition of three pointlike atoms in the crystal, these interactions are invariant with respect to their reflection in a plane that runs through the interacting atoms. Threeparticle interactions thus completely violate the Cauchy relations and maintain a certain relation between constants of thirdorder rigidity that does not correspond to crystal symmetry. For cubic crystals, this relation between the moduli of 3
thirdorder rigidity can be written as
C123 − 3C144 + 2C 456 = 0.
(4)
In modern theory, potential models in which the electron density and energy spectra can be calculated in the context of quantum theory are used to eliminate the Cauchy relations. The general approximation typ ical of all calculations based on quantum mechanical models is the selfconsistent field approximation, which results (no matter how the energy is calculated) in greater symmetry of the nonequilibrium potential than threeparticle interactions [14, 15]. There are thus relations between moduli of thirdorder rigidity, calculated on the basis of quantum mechanical models or a Hamiltonian that considers interactions in clus ters consisting of no more than three atoms, that can not be valid for any type of crystal over a wide range of variation in external conditions. Including irreducible interactions in clusters con sisting of no more than three atoms in the total elastic energy also leaves relations between the moduli of 3
Equation (4) was obtained in [12] for the partial potential of threeparticle interaction, ignoring invariance with respect to particle permutation and assuming that the potential depends only on the moduli rik. We can verify that Equation (4) is valid for crystals with arbitrary symmetry by using the IRBI of an O(3) ⊗ P3 group generated on the (rik) vectors that join the atoms in the triplets [11, 13].
fourthorder rigidity that do not correspond to sym metry and can be written for cubic crystals as (5) C1123 + 2C1456 − C1144 − 2C1244 = 0. It is clear from the above that we cannot in princi ple calculate all independent moduli of third and fourthorder rigidity without considering fourparticle interactions. In this work, we propose a method for calculating moduli of rigidity for pure metals and disordered alloys, based on presenting the elastic energy of a crys tal (ε) in the form of functions that are polynomials of the vector components ( rik ) that make up the IRBI for the group Gn = O(3) ⊗ Pn. Here rik are vectors that join different atoms of an alloy or metal, and n is the num ber of particles in the investigated cluster. When n = 2, G2 = O(3) ⊗ P2 ≡ C∞v; when n = 3, G3 = O(3) ⊗ P3, …. Group О(3) corresponds to when the energy of inter action of, e.g., three particles does not depend on the triplet’s position in space, ignoring interaction between atoms that are not part of the triplet; i.e., it is maintained with the triplet’s free rotation in space. The Pn group is the group of the permutation of the number of atoms in the cluster. In this first work, we only show how to calculate the moduli of second and thirdorder rigidity for copper at low temperatures by considering interaction in clusters that contain only pairs and triplets of atoms. It is very cumbersome to consider the energy of interaction in clusters that con tain quadruplets of atoms, and it could therefore be difficult to understand all features of our method for calculating moduli of highorder rigidity. For the same reason, questions of how to overcome restrictions on the calculation results associated with the possibility of relying on rational dependences ε 2 ( rik ) will be dis cussed in a separate work. This work is structured as follows. In the first part, we present the polynomials that form the integral rational basis of invariants for the group O(3) ⊗ P3, which are treated as rik vector functions. In the second part, we present the simplest models that consider the threeparticle interactions in an А1 structure on the basis of IRBI and discuss one possible way of establish ing a hierarchy for threeatom interactions in a partic ular structure, e.g., one typical of crystals of noble metals (Au, Ag, Cu, Pt). In the third part, we accept the simplest hypothesis on the possible dependence of the total energy of interaction for pairs and triplets of particles in the A1 structure. In this work, the aim of which is to illustrate the possibilities of our method for calculating moduli of secondorder (and higher) rigid ity, we use the simplest possible relation between the potential of interaction and invariants, which is deter mined by three phenomenological parameters. In part four, these parameters are found using the isothermal values for moduli of secondorder rigidity. With the obtained values for the potential parameters, we calcu late the relation between total energy and displace
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ment, along with moduli of rigidity predicted on the basis of nine model potentials written as the sum of two and threeatom interaction. In the final part, our results are compared with the results from calculations and experiments in the literature. INTEGRAL RATIONAL BASIS OF INVARIANTS (IRBI) Ignoring the calculations in [11, 13], we write the integral rational basis of invariants of the O(3) ⊗ P3 group generated on rik , vector components that join the atoms in a cluster containing three atoms with the numbers i, j, k: I 1 = rik2 + rkl2 + rli2, I 2 = rik rkl + rkl rli + rli rik , 22 22 22 I 3 = rik rkl + rkl rli + rli rik , I 4 = ( rik rkl )( rkl rli ) + ( rkl rli )( rli rik ) + ( rli rik )( rik rkl ) , I 5 = rik2 [( rik rkl ) + ( rli rik )] + rkl2 [( rik rkl ) + ( rkl rli )] (6) + rli2 [( rli rik ) + ( rkl rli )] , 222 I 6 = rik rkl rli , I 7 = ( rik rkl )( rkl rli )( rli rik ) , 2 2 I 8 = rik2 [( rik rkl ) + ( rli rik )] + rkl2 [( rik rkl ) + ( rkl rli )]
( )
2 + rli
( )
( ) [( r r ) + ( r r )], 2
li ik
kl li
2 2 2 2 I 9 = rik2 ⎡⎣( rik rkl ) + ( rli rik ) ⎤⎦ + rkl2 ⎡⎣( rik rkl ) + ( rkl rli ) ⎤⎦ 2 2 + rli2 ⎡⎣( rli rik ) + ( rkl rli ) ⎤⎦ . Hierarchy of ThreeParticle Interaction in an A1 Structure The crystal structure of noble metals (Cu, Au, Ag, Pt) is A1; the symmetry group is Oh5; and an elementary cell contains four primitives. To obtain the relation between the nonequilibrium potential and the basic invariants (6), we must first hypothesize as to the form of the relation between the particles’ energy of interac tion and their disposition in the lattice. Effective pair interactions between pointlike particles can depend only on rik moduli, since they are invariant with respect to the C∞V group [8]. Let us use the Lennard– Jones potential as a model for the potential of pair interactions: ε 2 ( rik ) = − A1rik−6 + A2rik−12, where А1 and А2 are model parameters that can be calculated by com paring the expressions for constants of secondorder rigidity. The values of the Lennard–Jones potentials for all metals were presented in [16, 17]. They were obtained by comparing the heat and elastic character istics for different crystals of the Periodic Table, calcu lated using ε2, with the same characteristics obtained experimentally. Since we use a more complete expres sion for the energy of interaction between particles that considers the contribution from threeparticle
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interaction to establish the parameters of pair interac tion potential, we must calculate А1 and А2 separately for each type of threeparticle interaction. Note that we can use the Van der Waals, Ridberg, or Morse potentials instead of the Lennard–Jones potential without substantially influencing the final results of our calculations [18, 19]. The surplus energy (relative to the energy of two atom interaction) of a cluster containing three atoms is generally a complex function of polynomials (6) In this work, we use the simplest version of such a rela tion: ε3 j = Q j I −j n, where Qj is a phenomenological parameter similar to A1 and A2 in the Lennard–Jones model of pair interaction potential. Parameter n is chosen such that the energy of interaction falls with the distance between any pair of atoms from an inter acting triplet, or faster than the energy of pair interac tion falls (rik−6; i.e., n ≥ 6). This is logical, since if one atom in a triplet is quite far from the other two, the potential energy of such a triplet is approximately equal to the energy of interaction of the pair formed by the atoms that are closest together. To calculate the values of invariants (6) at the nodes of an A1 structure [13], we must introduce a hierarchy of interaction for triplets of atoms similar to that of pair interaction. In phenomenological theory, a hierarchy of pair interac tion is established by calculating the number of coor dination spheres in which atoms are located relative to one another in a crystal lattice. Such a classification is impossible for triplets of atoms, and we therefore pro pose a generalized classification based on coordinate spheres. Let us examine our crystal lattice. The points of equilibrium position for the atoms that form an inter acting triplet we take to be the vertices of a triangle. The following can be used as a criterion allowing us to range triplets of interacting atoms in the order of diminishing triplet surplus energy moduli (i.e., the energies of triplets added to the sum of the energies of interacting pairs of atoms in a triplet): Let us compare the energies of interaction for the atoms that form the vertices of two triangles. Let the length of one side of the first triangle be equal to the length of a side of the second triangle. As a first approximation, we accept that the lower irreducible surplus energy of interaction corresponds to a triplet of atoms in which there is a tri angle where one side is longer than any of the sides of another triangle. To establish the hierarchy of total energies of the interaction of triplets of atoms, we mentally separate them into groups, each of which corresponds to a triangle equivalent to one of those listed in the second column of Table 1. To calculate the total energy of each group, we must consider the values of the phenomenological parameters that determine the relation between the energy and the invariants, and the number of identical triangles in the structure that contribute to invariant value (6), on which the energy of the triplets depends. We must of course hypothesize
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Table 1. Values of invariants I1, …, I9 from (6) that correspond to different types of interacting triplets found at lattice nodes А1 No. 1 2 3 4 5 6 7 8 9 10 11 12
Number of at oms in a triplet
Number of triplets
1,2,3 1,2,4 1,3,5 1,2,6 1,2,7 1,5,7 1,5,8 1,2,9 1,6,8 1,5,10 1,6,7 1,6,9
24 36 72 72 72 72 24 18 144 36 24 24
I1
I2
I3
I4
I5
I6
I7
I8
I9
6
–3 –4 –5 –6 –7 –8 –9 –6 –8 –8 –10 –12
12 20 28 44 60 84 108 36 76 80 132 192
3 4 3 8 11 20 27 0 12 16 32 48
–12 –24 –44 –56 –76 –88 –108 –72 –104 –96 –136 –192
8 16 24 48 72 144 216 32 96 128 288 512
–1 0 9 0 –5 –16 –27 32 0 0 –32 –64
–24 –80 –232 –288 –440 –496 –648 –528 –736 –640 –944 –1536
12 48 148 192 316 272 324 336 544 384 496 768
8 10 12 14 16 18 12 16 16 20 24
as to the relation between the energy and invariants (6) before anything else. This is the necessary sequence of actions that allows us to construct an approximate hierarchy of the threeparticle interactions that must be considered the total energy of a given structure. The hierarchy of energies of threeparticle interaction is refined for each particular structure during calcula tions. Model Potentials of ThreeParticle Interaction in an А1 Structure with Symmetry Considered Table 1 gives an example of how to construct a hier archy of threeparticle interaction when atoms are located at the nodes of the crystal lattice of an А1 structure. The column headings are given in the first line. We assume that the minimum potential energy of irreducible threeatom interaction in an А1 structure corresponds to those triplets that we can place with the atoms numbered 1, 2, and 3 in the figure by hypothet ically shifting the entire triplet in space. Equilateral triangles with sides of 2τ correspond to such triplets. 9 10
7 5 3
Here τ denotes half the length of the edge of an ele mentary cell. There are 24 triangles that have one ver tex at node number one (see figure). This description allows us to understand the meaning of the numbers in the first three cells of the second line in Table 1. The numbers in cells 4, 5, …, 12 of the second line are the values of invariants 1, 2, 3, …, 9 from (6) generated on vectors r12, r23 and r31.. The numbers in lines 3, …, 13 in Table 1 have exactly the same meaning. The number of lines in Table 1 is determined by the limit on the value for the maximum side of the triangles considered in one model or another of the potential energy of atomic interaction. We restrict ourselves to considering the interaction of triplets of atoms, in which the maximum distance between two atoms is (2)3/2τ. It is easy to see that in an А1 structure, such a restriction can result in a relative error on the order of 2–(3p + 4)/2 in calculating the mod uli of secondorder rigidity, and of 2–(3p + 6)/2 for moduli of thirdorder rigidity. Here p = n ⋅ m, where m charac terizes the total degree of dependence of invariant number j on rik , in ε3j. The general procedure for cal culating the numerical values of the invariants is pre sented in [13]. The invariants presented in columns 4– 12 are related to the A1 structure. The numbers of atoms presented in the second column of Table 1 cor respond to the figure, which gives a clear picture of the disposition of atoms in a certain type of triplet in an A1 structure.
6 2
4 1
8
New method for considering threeparticle interaction in the theory of modulus elasticity.
in
∑
In the case of the potential of atom interaction used this work, ε (i, k, l j ) ε 2 + ε3 j = = −6 ⎡⎣− A1rik + A2rik−12 + Q j I −j n ⎤⎦, where i, k, and l surpass
i,k,l
the values corresponding to the number of atoms that
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form a triplet. In each of the nine models (j = 1, …, 9), the energy of the members of the expansion of total energy E with respect to the sums of the energies of identical clusters ε (i, k, l j )] depends on three phe nomenological parameters: A1, A2 and Qj. These parameters, as usual [1–3], can be calculated using the experimental values for elastic moduli of the second order: C11, C12, and C44 [20–23]. Since the phenome nological parameters differ for each of the models, all further calculations require that we specify the sub stance to be investigated. Only after determining the numerical values of the phenomenological parameters can we discuss the question of whether it is possible to limit our calcula tions by considering the interaction of three particles when the maximum distance between them is (2)3/2τ.
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Table 2. Values of the phenomenological parameters for nine models that consider threeparticle interactions. Pa rameters were calculated using the moduli of secondorder Exp elasticity for monocrystal copper known from [20]: C11 = Exp Exp = 1.199 × 102 GPa; C44 = 0.756 × 102 GPa 1.661 × 102; C12
А1
А2
Qj
j=1
0.283
0.918
1.54
j=2
0.283
0.918
–4.83 × 10–2
j=3
0.250
0.850
0.151
j=4
0.173
0.685
3.73 × 10–3
j=5
0.304
0.977
0.228
j=6
0.232
0.807
6.55 × 10–3
j=7
0.191
0.780
–1.48× 10–3
Moduli of ThirdOrder Rigidity for Monocrystal Cu
j=8
0.373
1.112
–4.69 × 10–2
According to [20], the experimental values for moduli of secondorder rigidity for Cu monocrystal are C11Exp = 1.661 × 102 GPa, C12Exp = 1.199 × 102 GPa,
j=9
0.429
1.237
2.93 × 10–2
4
Exp = 0.756 × 102 GPa. C 44 In other experimental and theoretical works where C11, C12 and C44 are presented for Cu crystals [21–23], the moduli of secondorder rigidity were taken from [20]. Phenomenological parameters A1, A2 and Qj, cal culated on the basis of these data for monocrystals of Cu, are presented in Table 2 for all nine models. As can be seen from these values, the parameters that deter mine the Lennard–Jones potential and the potential of threeparticle interaction differed considerably for different models. This shows that the introduced phe nomenological parameters of semiempirical pair interaction are conditional characteristics of a sub stance. Their values, like those of the phenomenolog ical parameters of other semiempirical potentials of pair interaction, correlate in particular with the assumed type of threeparticle interaction. This is never mentioned in works that give parameters deter mining pair interaction for crystals of the Periodic Table (e.g., in [7, 16, 17, 25]). It is the task of phenom enological models to ensure that the measurable char acteristics of a substance calculated within these mod els and the values obtained as a result of actual mea surements are in good agreement. It is according to this criterion that the best phenomenological model is determined. It is therefore necessary to compare the obtained experimental values for moduli of third order rigidity for copper and the values of the same moduli calculated in different models, including the nine models defined by the three parameters presented in Table 2.
4 Several
different values for the moduli of rigidity of Cu are pre sented in [24], but there are no references to the type of mea surements that produced such numerical values.
The results from calculations needed for such a comparison are presented in lines 2–9 of Table 3. The moduli of thirdorder rigidity enumerated in the table columns are given in the first line of Table 3. The experimental values for the moduli of third order rigidity obtained in [20–22] are presented in lines 11–14. Interference methods sensitive to small variations in the velocity of ultrasound were used for this purpose in [20]. We determined how the velocity varies due to the impact of uniaxial or hydrostatic pressure in order to calculate the moduli of thirdorder rigidity using the interference data. To determine six moduli of rigidity, we solved 14 equations in the theory of elasticity that determine the velocity of sound with different polar izations and directions of propagation along the axis of symmetry of a cubic crystal with different geometries of imposed external stresses. The measurements were conducted at room temperature. In [21], the measure ments were conducted at 295, 77, and 4.2 K. The results of both works differ from Milder’s formula. In [22], the results of [20–21] were subjected to careful verification at 295K and were found to lie within the indicated margin of error. Our calculations correspond to 0 K; i.e., they must be compared to the data the obtained at 4.2 K. Note that the data obtained in [21] at 4.2 K are accepted as the most reliable in [23], where they are used to calculate the phenomenologi cal parameters of the quantum mechanical model in an approximation of an immersed atom. It is also worth comparing the values for moduli of thirdorder rigidity obtained as a result of calculations performed with phenomenological models 1–9 and on the basis of the model (semiempirical) Hamilto nians used in contemporary approximate approaches
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Table 3. Moduli of thirdorder rigidity, calculated according to the phenomenological parameters for the models presented in Table 2. For comparison, we present the moduli of thirdorder rigidity for Cu measured in [20–22] and calculated in [1–3, 23] using different approximations of models in quantum mechanics Cu
C111
C112
C166
ε(i, k, l|1)
–18.1
–10.6
–10.4
–0.48
ε(i, k, l|2)
–20.9
–12.6
–10.9
–1.30
–0.27
0.24
ε(i, k, l|3)
–22.5
–13.5
–10.8
–2.23
–0.54
0.31
ε(i, k, l|4)
–18.7
–11.9
–2.35
–0.45
0.49
ε(i, k, l|5)
–25.6
–14.5
–11.9
–2.51
–0.62
0.33
ε(i, k, l|6)
–22.4
–13.7
–10.6
–2.49
–0.63
0.29
ε(i, k, l|7)
–19.2
–12.9
–1.99
0.10
1.14
ε(i, k, l|8)
–31.1
–16.3
–13.4
–3.18
–1.0
0.1
ε(i, k, l|9)
–34.6
–17.3
–14.6
–3.47
–1.05
0.16
–0.5 ± 0.18
–0.03 ± 0.09
–0.95 ± 0.87
–9.31
–9.95
–7.8 ± 0.05
C123
C144
C456
0.035
0.29
Exp [20], 295 K
–12.7 ± 0.22
–8.14 ± 0.09
Exp [21], 295 K
–15.0 ± 1.5
–8.5 ± 1.0
–6.45 ± 0.1
–2.5 ± 1.0
–1.35 ± 0.15
–0.16 ± 0.1
Exp [22], 295 K
–13.7 ± 0.73
–8.19 ± 0.23
–7.24 ± 0.2
–1.59 ± 1.14
–0.64 ± 0.62
–0.11 ± 0.88
Exp [21,23], 4 K
–20 ± 2
–12.2 ± 1.5
–7.05 ± 0.25
Calc [1], L PDP
–17.0
–9.65
–8.32
Calc [2], L PDP
–11.9
–6.46
–8
Calc [3], DFT
–15.07
–9.65
–9.01
within quantum mechanics. Moduli of thirdorder rigidity for metallic copper were calculated by means of quantum mechanics in [1–3]. In [1], moduli of thirdorder rigidity were deter mined as derivatives of energy density with respect to tensor components of uniform deformations of the total energy, which included four components: free electron energy, electrostatic energy, the energy of overlapping ion core shells, and the energy of the zone structure. The energy of the zone structure was expressed in terms of local pseudopotential. The energy of overlapping ion core shells was approxi mated by theBorn–Maier potential. After considering that a partial type of the local pseudopotential was used in [1], the total energy was determined by five fit ting parameters found using the experimental values for binding energy, interatomic distances, and three moduli of secondorder rigidity. From the values obtained in this manner for the filling parameters, it became clear that the energy of overlapping ion core shells makes the main contribution to the value of moduli of thirdorder rigidity. The contributions from other summands of the total energy are not negligible, but they compensate for one another to a considerable degree. In [2], similar calculations were performed in the same approximation, but moduli of third and fourthorder rigidity were calculated.
–5 ± 1.5
–1.32 ± 0.2
0.25 ± 0.08
–0.10
0.34
0.12
2.19
0.17
0.01
–0.71
–0.07
0.45
A first principles method for calculating in an approximation of the electron density functional was used in a later work by Wang and Li [3]. Attention was focused mainly on the similarity of the results from calculating the values of internal energy and the mod uli of rigidity if the size of the grid cell points is reduced in k space. The phenomenological parameters of elec tron density in [3] were taken from [26]. RESULTS AND DISCUSSION From the beginning, we have viewed our calcula tion results as an example illustrating the possibilities of the proposed approach based on IRBI. As is well known, IRBI reflects most fully the properties of mul tiparticle interactions, which are determined by the symmetry of a problem. The illustrative nature of our calculations is due to fourparticle interactions, which are not reducible to pair or triplet interactions, not being considered in this work. It thus follows, in agree ment with Eq. (4), that not all of our calculated moduli of thirdorder rigidity are independent. Note especially that we use three fitting (phenom enological) parameters in each of the nine models examined in this work. In first principle calculations (e.g., in the pseudopotential method [1, 2, 27, 28]), a minimum of five fitting parameters are used, two of which determine the arbitrarily selected potential of
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pair interactions (e.g., the Born–Maier potential [1, 2, 27]). In the immersed atom method, the number of fitting parameters is sometimes more than 20 (due to the type of interatom interactions being speci fied [3]). Let us compare the results from calculations per formed using the different procedures for obtaining moduli of thirdorder rigidity presented in this work and in [1–3] with the results obtained by measuring changes in the velocities of polarized ultrasound with different geometries of its propagation and different directions of imposed uniaxial stresses [20–22]. Remember that the experimental results are neither adiabatic nor isothermal values for moduli of rigidity. The respective corrections for С111 and С112 are rela tively small, but for the shear moduli (С144, С456) of plastic metals, they can exceed the error of the mea surements [22]. In addition, the mixed character of the experimental moduli results in apparent violation of external symmetry: the measured values С144 ≠ С441 for, e.g., cubic crystals. Note too that the features of acoustic contacts mean the C111, C112, and C123 moduli are measured with greater accuracy than moduli containing shear indexes (4, 5, 6). Except for j = 8, 9, the numerical values of the ten sor components of copper moduli of rigidity calcu lated on the basis of the proposed models ε (i, k, l j ) = ⎡⎣− A1rik−6 + A2rik−12 + Q j I −j n ⎤⎦, are in good agreement
∑ i,k,l
5
with the measurement results [21]. We can see from a comparison of the calculation results that model 4 is the one closest to the experimental data. The results from calculations performed by means of quantum mechanical models do not show better agreement between the calculated and experimentally established values, even though a greater number of phenomeno logical (fitting) parameters are used. ACKNOWLEDGMENTS We thank A.Yu. Gufan for participating in discus sions of our results. This work was supported by the Russian Foundation for Basic Research, project nos. 100200826a and 100500258a. REFERENCES 1. Thomas, J.F., Phys. Rev. B, 1973, vol. 73, no. 6, p. 2385. 2. Soma, H. and Hiki, Y., J. Phys. Soc. Jpn., 1974, vol. 37, p. 544. 5 In
today’s literature, values for moduli of secondorder rigidity are sometimes taken from [24]. These values differ somewhat from the values we used in calculating the phenomenological parameters for the models presented in Table 2. Using the data in [24] changes the results presented in Table 4 only slightly.
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BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS
Vol. 75
No. 9
2011
