fulltext

Three-body effect on the lattice dynamics

of some fcc d-band metals

I.. AKGU¨N, G. UGˇUR, A. GU¨NENand M. C¸I.VI.

Gazi University, Faculty of Arts and Sciences - Ankara, Türkiye (ricevuto il 9 Dicembre 1996; approvato il 25 Febbraio 1997)

Summary. — In the present analysis, the interaction system of an fcc d-band metal is considered to be composed of two-body and three-body parts. We use a new three-body potential developed by Akgün and Ugˇur to deduce the contribution of many-body forces to the dynamical matrix of the fcc structure. Two- and three-body potentials are first used, as an application to investigate the dynamical behaviors of fcc d-band metals, Ni, Pd, Cu and Ag. The parameters defining the two- and three-body potentials for the metals are evaluated from knowledge of the equilibrium pair energies, bulk modulus and total cohesive energies of the metals, following a procedure given by Akgün and Ugˇur. In this scheme the input data is independent of phonon frequencies and elastic constants of the metals. Finally, the phonon frequencies of the metals along the principal symmetry directions are computed using the calculated radial, tangential and three-body force constants. The theoretical results are compared to experimental phonon dispersions. The agreement shows that the proposed potentials and crystal model provide a reasonable description of the lattice dynamics of fcc d-band metals.

PACS 63.20 – Phonons in crystal lattices.

1. – Introduction

Interatomic pair potentials have long been used in investigating the lattice, elastic, and electronic properties of (d-band noble and transition) metals. A formal derivation of the total energy of a solid in terms of N-body interactions can be achieved in many ways. In both the embedded-atom model (EAM) [1] and the Finnis-Sinclair model [2] which are known as a new approach the total energy is expressed as a sum of pair potentials that are explicit functions of the interatomic distance and atomic energies, the so-called “embedded atom” terms. In the EAM, the total energy of a homonuclear arrangement of atoms is given by

Etot4

!

!

accounts for the cohesive energy of the solid. F(rh , i) is obtained by summing over the

free atom densities calculated by ab initio methods. In the Finnis-Sinclair scheme [2], the electron density at the nucleus is made an explicit function of the distances to the neighbouring nuclei. The many-body feature of these schemes arises by making the atom energy F(rh , i), a nonlinear function of rh , i. An approximation for the electron

density rh at atom is provided by a linear superposition of all other atomic electron

densities [3]:

rh , i4

!

j c i ra(rij) ,

(2)

where ra_{(r) is the atomic electron density. In eq. (1), the pairwise term takes the form}

f(rij) 4 Z2_(r ij) rij , (3)

where the effective charge Z(rij) is a function of atomic species and interatomic

distance rij. There have been several variations of these schemes with different

approaches to parameterization [3-5]. A popular devolopment is that of Johnson [4] who defines F(rh , i) from a universal curve of energy vs. distance proposed by Rose et al. [6]

It should be noted that the pair potentials in these schemes are short range, typically not extending beyond second neighbours. The predictions of the EAM are tested by Daw and Hatcher [3], by the calculation of phonon dispersion functions in Ni and Pd. They used the functions F(rh , i), and f(rij) derived by Daw and Baskes [1] for Ni and

Pd. These functions were determined by fitting to properties of the bulk metals [1]: lattice constant, elastic constants, cohesive energy, vacancy-formation energy, and difference in body-centred-cube (bcc) and face-centred-cube (fcc) phase energies. The results obtained by Daw and Hatcher [3] for Ni and Pd are represented in figs. 1, 2, respectively. To calculate the phonon dispersions in Cu, the pair potentials and the volume-dependent energies in the form of the embedding energy were also calculated by using the interplanar force-constant formula connected with the functions F(rh , i)

and f(rij) in the EAM [5]. In this work [5], the functions F(rh , i) and f(rij) for Cu have

been derived in the same way that Daw and Baskes [1] derived them for Ni and Pd. The phonon dispersion curves obtained for Cu are represented in fig. 3.

The EAM models are quite succesful for some transition metals as well as simple metals but they are not generally used for covalent solids. The question still to be answered is whether there is a common strategy that can be applied to atomic solids regardless of the nature of their bonding. Theoretical models for lattice dynamical studies of metals have usually been tested by comparing the phonon frequencies calculated by emprical many-body potentials with those measured experimentally.

In the present paper, the interaction system is composed of two-body and three-body parts. Recently, a two-body potential has been developed by Singh and Rathore [7] and it has only been tested for phonon dispersion curves of fcc Fe. To include the most significant contribution to the binding energy, which arises from the interaction between the metal ions and electrons, a new three-body potential based on the two-body model potential [7] has been developed by Akgün and Ugˇur [8] and it has successfully been applied to the phonon frequencies of the fcc Pd-10%Fe alloy [8].

Fig. 1. – Phonon dispersion curves of Ni at room temperature. The points are experimental data by Birgeneau et al. [19]. The solid curves show the dispersion curves computed by including the contribution of three-body forces and the dashed curves represent the computed dispersion curves according to the two-body central interactions. The dashed-dot curves represent the calculations by the EAM.

In the present work, the two-body atomic interactions are given by [7]

f2(rj) 4

!

D

2(m 21)rj

[

exp [2marj] 2mb exp [2arj]

]

(4)

Fig. 2. – Phonon dispersion curves of Pd at room temperature. The points are experimental data by Miiller et al. [20]. The rest of the description is the same as that of fig. 1.

Fig. 3. – Phonon dispersion curves of Cu at room temperature. The points are experimental data by Svensson et al. [21]. The rest of the description is the same as that of fig. 1.

and the three-body interactions are given by [8]

f3(r1r2) 4

!

!

exp [2ma(r11 r2) ] 2mb exp [2a(r11 r2) ]( ,

where r1and r2are the respective separations of the atoms (l 8, k 8) and (l 9, k 9) from

the atom (l , k). C is the only parameter in the three-body potential to be evaluated. In eq. (4), D is the dissociation energy of the pair, a is the constant which measures the hardness of the potential, m is an exponent which delivers the same effect to the potential as results from the exchange and correlation effects due to electrons, b 4 exp [ar0], r0 is the separation of the atoms for minimum potential and the term rj21

modifies the potential to exhibit the correct nature of the forces. The distance of the

j-th atom from the origin rj4 a(mj21 nj21 lj2)1 /2, where mj, nj, lj are integers

representing the coordinates of the j-th atom of the lattice and a is the lattice constant. Therefore, the aim of the present work is to show that the scheme expressed in sect. 2 and the proposed potentials (4), (5) provide a reasonable description to the problem of studying the lattice dynamics of fcc d-band metals, Ni, Pd, Cu and Ag.

2. – Theory and computation

The total interaction energy of a system of N atoms, in general, may be expressed as a many-body expansion,

f 4f21 f31 R 1 fn1 R ,

(6)

where f2, f3and fnrepresent the total two-body, three-body and n-body interaction

TABLEI. – Input data.

Metal a ( nm ) ref. 2e0(eV) ref. B ( 1011N/m2) ref. 2f ( eV ) ref.

Ni 3.52 [11] 2.07 [9] 1.86 [11] 4.44 [11]

Pd 3.89 [11] 1.10 [10] 1.80 [11] 3.89 [11]

Cu 3.61 [11] 2.01 [9] 1.37 [11] 3.49 [11]

Ag 4.09 [11] 1.65 [9] 1.00 [11] 2.95 [11]

system simply by separating C as

f 4f21 Cf3,

(7)

where f is taken as bulk cohesive energy per atom and C is the only parameter in the three-body potential to be evaluated. The two-body interaction energy, f2, is given by

eq. (4). The parameters (a, r0, D) defining the pair potential (4) for fcc d-band metals,

Ni, Pd, Cu and Ag, are computed following the procedures given by Akgün and Ugˇur [8]: at equilibrium semilattice constants of the metals, a0,

.

`

/

`

´

Nr 4a

Nr 4a

where e0is the pair energy at equilibrium, i.e. e0is the ionic part of the bulk cohesive

energy, B is the bulk modulus, and c is a geometrical constant depending on the type of the crystal (for fcc crystal c 42). For the metals the input data used in eqs. (7), (8) are given in table I.

In the computations we have considered the couplings extending to the eighth neighbor of the fcc structure. Eight forms of the pair potential (4) obtained by varying the exponent m as 1.01, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0 are studied for the metals, i.e. the potential parameters (D , a , r0) are evaluated for each value of m, separately. In order

to determine the best values of the exponent m defining the two-body potential (4) for the metals, we have then computed the elastic constants (C11, C12, C44) of the metals,

separately. The elastic constants can be evaluated from the well-known expressions for cubic crystals with two-body interatomic interactions [12, 13]:

.

`

/

`

´

!

!

C11 C12 C44 C11 C12 C44

Ni 2.00 2.50 1.53 1.15 2.61 1.50 1.23

Pd 2.50 2.42 1.51 1.11 2.34 1.76 0.71

Cu 1.01 1.79 1.16 0.80 1.76 1.24 0.81

Ag 1.01 1.31 0.87 0.58 1.31 0.97 0.51

TABLEIII. – Computed two-body and three-body (C) potential parameters. Metal m D ( 10229_{Jm) a ( 10}10_m21_{) r} 0 ( 10210m) C Ni 2.00 1.154667 1.803466 2.617913 0.0207 Pd 2.50 7.701210 2.427558 2.785278 0.0648 Cu 1.01 1.144502 2.246871 2.720622 0.0512 Ag 1.01 1.130709 2.310210 3.014504 0.0769

devoloped by Milstein and Rasky [13] is used because they have noted that the relations in eq. (9) are in better agreement with experimental data than the Cauchy relation

C444 C12 for fcc crystals. Also, using C444 ( 1 O3 )( 2 C112 C12), the elastic constants of

the fcc Fe-35%Ni alloy have been computed by Akgün [14], and it is seen that the obtained results provide theoretical support for the efficacy of this relation in the fcc alloys. Thus the elastic constants of the metals are calculated separately from eq. (9) for the values m given above. Comparing the calculated values with experimental values of the elastic constants we have determined the values m given in table II for the metals, separately. For the determined values of the exponent m, the computed parameters (a, r0, D) of the two-body potential (4) are given in table III. The

three-body interaction energy, f3, is given by eq. (5). For the three-body interaction

considered here, the first neighbor of the fcc configuration is regarded as common nearest neighbor of the second and third neighbors. Thus the three-body potential parameter, C, in eq. (5) can be evaluated easily by fitting the total interaction energy of an atom in a particular crystal structure to the total cohesive energy f of the metal. For the metals the values of the three-body potential parameter computed from eq. (7) are given in table III.

In the harmonic and adiabatic approximations, the phonon frequencies corres-ponding to a wave vector kK for a cubic crystal are determined by solving the secular equation, given by

ND 2 MW2IN40 ,

(10)

where D is the dynamical matrix of order ( 3 33), I is the unit matrix, and M is the ionic mass. In the present work, the elements of the dynamical matrix Dabare composed of

two-body (Di

ab) and three-body (Dabm) parts:

Dab4 Dabi 1 Dabm.

TABLEIV. – Computed radial (ai) force constants. Serial no. ai ( 1023Nm21) Ni Pd Cu Ag 1 2 3 4 5 6 7 8 30523.6 21897.29 2486.310 2130.013 240.0478 213.8638 25.25885 22.14511 27693.7 2788.841 277.1575 210.9295 21.98976 20.43179 20.10684 20.02929 24612.7 21726.59 2525.617 2141.924 241.4005 213.1466 24.49851 21.64062 18725.1 21288.64 2267.410 254.9561 212.7091 23.28081 20.92908 20.28432

TABLEV. – Computed tangential (bi) force constants.

Serial no. bi ( 1023Nm21) Ni Pd Cu Ag 1 2 3 4 5 6 7 8 2761.11 312.317 57.0165 13.0587 3.61413 1.14967 0.40624 0.15583 2117.81 76.6107 6.10768 0.75776 0.12440 0.02479 0.00570 0.00146 2787.08 316.382 61.4316 13.5019 3.42730 0.97843 0.30698 0.10403 2370.62 185.115 26.0750 4.41707 0.89346 0.20792 0.05408 0.01539

In the case of the two-body central interaction, the interactions are assumed to be effective up to eight nearest neighbors and the Di

ab are evaluated by the scheme of

Shyam, Upadhyaya, and Upadhyaya [16]. The typical diagonal and off-diagonal matrix elements of Di

abcan be found in ref. [16]. In the case of the central interaction, first and

second derivatives of the two-body potential f2(r) provide two independent force

constants, i.e. the radial force constant aiand tangential force constant bi, for the i-th

set of neighbors:

.

`

/

`

´

Nr 4r

Nr 4r

For the metals the computed force constants are given in table IV, V.

In order to determine the contribution of the three-body forces to the diagonal and off-diagonal matrix elements of Dm

ab, we follow the scheme of Mishra et al. [17], where a

. / ´ Daa4 4 g[ 4 2 2 C2 i2 Ci(Cj1 Ck) ] , Dm ab4 4 g[Ci(Cj1 Ck) 22] , (13)

where g is the second derivative of the three-body potential f3(r1r2), Ci4 cos (paki)

and C2 i4 cos ( 2 paki). To calculate the three-body force constant g, we limit the

short-range three-body forces in the fcc system only up to first-nearest neighbors. The computed values of the three-body force constant are g 40.631 Nm21 _{for Ni, g 4}

1.795 Nm21 _{for Pd, g 41.260 Nm}21 _{for Cu, and g 41.441 Nm}21 _{for Ag. Thus one can}

construct the dynamical matrix Dab by using eq. (11) and then solve the secular

equation (10) to compute the phonon frequencies along the principal symmetry directions [100], [110], and [111] for the metals.

3. – Results and discussion

In the present analysis, the interaction system of fcc d-band metals, Ni, Pd, Cu, and Ag, is considered to be composed of the two-body and three-body parts. By the three-body interaction we mean an extra interaction energy owing to the presence of the third particle. This type of interaction may occur through the deformation of the electron shells [18]. The two-body (4) and three-body (5) model potentials are first used, as an application, to investigate the lattice dynamics of the fcc d-band metals. In the case of the two-body interaction, we have considered couplings extending to the eighth neighbor of the fcc structure. The parameters (a , r0, D) defining the model

potential f2(r) for the metals are evaluated for the fcc structure, by the knowledge of

the equilibrium pair energies and bulk modulus of the metals. The three-body parameter C is evaluated from the knowledge of the total cohesive energies of the metals. Thus, on the one hand, we determine the ab initio radial (ai) and tangential

(bi) force constants of metals by using the model potential (4) and, on the other, we

reasonably account the long-range character of interatomic forces by considering the interaction system extending up to eight sets of nearest neighbors. The three-body force constants (g) of the metals are calculated by using the three-body potential (5). The computed values of g will depend on the relative magnitude of the repulsive and attractive three-body forces.

First, the phonon dispersion curves of the metals are only computed according to the two-body central interaction and the results are shown by dashed curves in figs. 1-4. Next, the calculations are repeated for the metals, by including the contribution of three-body forces to the dynamical matrix (11) and the computed dispersion curves are shown by solid curves in figs. 1-4, together with the corresponding experimental points for comparison. As seen from figs. 1-4, the experimental and theoretical values are in good agreement when the three-body forces are incorporated in the metals. According to this result the input data being independent of phonon frequencies and elastic constants enhance the reliability and credibility of the present analysis.

Furthermore, for Ni, Pd and Cu the phonon dispersion curves calculated by the EAM [3, 5] are represented in figs. 1-3, respectively. As seen from figs. 1-2, the agreement between the experimental findings with the results calculated by the EAM

Fig. 4. – Phonon dispersion curves of Ag at room temperature. The points are experimental data by Kamitakahara et al. [22]. The rest of the description is the same as that of fig. 1. There are not the calculations by the EAM for Ag.

for Ni and Pd is good for small qKin the symmetry directions but it is not for large qK, of course, because the functions F(r) and f(r) were determined by fitting to the elastic constants. However, the results calculated by the EAM for Cu are in excellent agreement with the measurements of the symmetry phonon frequencies (visible in fig. 3). Recently, Animalu [23] has developed a transition metal model potential (TMMP) of the Heine-Abarenkov type [24] and applied it to calculation of the phonon dispersion of some fcc d-band metals, Cu, Ag, Ni and Pd. We observe that, in general, the transverse branches are not in good agreement with the neutron data. It is to be pointed out that the TMMP of Animalu on the Fermi surface approximation is a local pseudopotential and describes in second order the interaction in the central pairwise form [25].

Consequently, the close scrutiny of figs. 1-4 shows that the proposed two- and three-body potentials are sufficient to reproduce the phonon data and the scheme expressed in sect. 2 works well for the fcc d-band metals.

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