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Three-body effect on the lattice dynamics of Fe-28%Pd alloy (*)

I.. AKGU¨Nand G. UGˇUR

Gazi University, Faculty of Arts and Sciences - Ankara, Turkey

(ricevuto l’11 Ottobre 1996; approvato il 27 Gennaio 1997)

Summary. — In the present work, a recently developed three-body potential and a new two-body model potential are first used, as an application, to investigate the dynamical behaviour of fcc Fe-28%Pd alloy. For this purpose, two- and three-body interactions have been employed to devolop the dynamical matrix of fcc structure. The parameters defining the two- and three-body potentials for Fe and Pd have been computed following a new procedure described by Akgün and Ugˇur. In this scheme, the input data for evaluating the necessary parameters is independent of the phonon frequencies and elastic constants of the alloy and metals. The force constants of the alloy have been calculated by the concentration averages of the computed force constants of the component metals. Finally, the phonon frequencies of the alloy along the principal symmetry directions have been calculated from the elements of the matrix so developed. The theoretical results have been found to be in good agreement with the corresponding experimental values.

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 metals and alloys. Furthermore, the contributory role of the many-body forces in determining the above properties of the metals has been emphasized by several workers [1-6]. While many different estimates of the three-body force [7-10] have been proposed, the two-body interaction has usually been described in terms of the derivatives of an unknown interionic potential.

In the present paper, it is considered that the interaction system is composed of two-body and three-body parts. Recently, a two-body potential has been developed by Singh and Rathore [11] and it has 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 [11] has been developed by Akgün and Ugˇur [12] and

(*) The authors of this paper have agreed to not receive the proofs for correction.

it has only been applied to the phonon frequencies of a fcc alloy. To investigate the dynamical behaviour of fcc Fe-28%Pd alloy, the two-body atomic interactions are given by [11] f2(rj) 4

!

[

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

]

and the three-body interactions are given by [12] (2) f3(r1r2) 4

4

!

l 8k 8cl 9k 9

!

CD

2(m 21)(r11 r2)

]bmexp [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. (1), 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, r0is the

separation of the atoms for minimum potential, b 4exp [ar0] and the term rj21modifies 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.

The interesting feature of Fe-Pd Invar alloys is that the two constituents of the alloy are in different phases, as Fe is in bcc phase and Pd in fcc phase at room temperature. However, the alloy Fe-28%Pd forms a random solution having a fcc structure [13]. Due to their structural complexes, not much theoretical work has been done on Fe-Pd alloys, to reveal the lattice dynamics. Therefore, the purpose of the present work is to show that the scheme expressed in sect. 2 and the proposed two- and three-body potentials (1), (2) provide a reasonable description to the problem of studying the lattice dynamics of fcc binary type-II alloys.

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 , (3)

where f2, f3and fnrepresent the total two-body, three-body and n-body interaction energies, respectively. In this paper we re-expressed the total interaction energy of a system simply by separating C as

f 4f21 Cf3,

(4)

where f is taken as total cohesive energy per atom and C is the only parameter in the three-body potential (2) to be evaluated. The two-body interaction energy, f2, is given

by eq. (1). The parameters (a , r0, D) defining the pair potential (1) for Fe and Pd are

computed for fcc structure at the lattice constant of the Fe-28%Pd alloy, following the procedure given by Akgün and Ugˇur [12]. The lattice constant of the alloy is

a 40.375 nm. At equilibrium semilattice constant of the alloy, a0,

.

`

/

`

´

N

N

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

energy, and k is the force constant at equilibrium. These parameters (e0, k) are

available in the literature for most metals. For Fe and Pd the input data used in eqs. (4), (5) are given in table I, where f is the total cohesive energy.

In the computations we have considered the couplings extending to the eighth neighbor of the fcc structure. Eight forms of the pair potential (1) obtained by varying the exponent m as 1.01, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, and 5.0 are studied for Fe and Pd, 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 (1) for the elements, we have then computed the elastic constants (C11, C12, C44) for the

fcc structure at the lattice constant of the alloy. The elastic constants can be evaluated from the well-known expressions for cubic crystals with two-body interatomic interactions [17, 18]:

.

`

/

`

´

!

!

, V is the volume per atom. For C44, the theoretical expression

devoloped by Milstein and Rasky [19] is used because they have noted that the relations in eqs. (6) are in better agreement with experimental data than the Cauchy relation C444 C12for fcc crystals. Also, using C444

1

3( 2 C112 C12), the elastic constants

of the fcc Fe-35%Ni alloy have been computed by Akgün [20], and it is seen that the results obtained provide theoretical support for the efficacy of this relation in the fcc TABLEI. – Input data for Fe and Pd from refs. [14-16].

Element 2e0( eV ) k ( eV/nm2) 2f ( eV )

Fe 0.90 926 4.28

TABLE II. – Computed elastic constants (in units 1011 _N/m2_{) for Fe and Pd at room}

temper-ature.

Element m Present work Experimental [9, 10]

C11 C12 C44 C11 C12 C44

Fe 3.5 2.43 1.31 1.18 2.30 1.35 1.17

Pd 1.5 2.60 1.42 1.26 2.27 1.76 0.71

TABLEIII. – Computed two-body and three-body (C) potential parameters for Fe and Pd at the

lattice constant of alloy.

Element m D ( 10229_{Jm) a ( 10}10_m21_{) r}

0( 10210m ) C

Fe 3.5 5.94161 2.12330 2.69146 18.0678

Pd 1.5 7.53246 3.09911 2.69027 84.5466

alloys. Thus the elastic constants of Fe and Pd at the lattice constant of Fe-28%Pd alloy are calculated separately from eqs. (6) 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 Fe and Pd. For the determined values of the exponent

m, the computed parameters (a , r0, D) of the two-body potential (1) are given in

table III. The three-body interaction energy, f3, is given by eq. (2). For the three-body

interaction considered here, the first neighbor of the fcc configuration is regarded as the common nearest neighbor of the second and third neighbors. For Fe and Pd, the values of the three-body potential parameters computed from eq. (4) at the lattice constant of the alloy Fe-28%Pd are given in table III.

The secular determinant to determine the frequency of vibration of a solid is given by ND 2 MW2IN40 ,

(7)

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 central pairwise (Di

ab) and three-body (Dabm) parts:

Dab4 Dabi 1 Dabm. (8)

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

ab are evaluated by the scheme of Shyam et al. [21]. The typical diagonal and off-diagonal matrix elements of Di

ab can be found in ref. [21]. In the case of the central interaction, first and second derivatives of the two-body potential f2(rj) provide two independent force constants, i.e. the radial

force constant ajand tangential force constant bj, for the j-th set of neighbors:

.

`

/

`

´

For Fe and Pd, the calculations of aj and bj are done for fcc structure at the lattice constant of the alloy Fe-28%Pd. Now we evaluate the force constants (ajand bj) of the alloy by using the linear relations

. / ´ aj( Fe-Pd ) 4 (12X) aj( Fe ) 1Xaj( Pd ) , bj( Fe-Pd ) 4 (12X) bj( Fe ) 1Xbj( Pd ) , (10)

where X is the concentration of Pd in the alloy (X 40.28). For Fe, Pd and Fe-28%Pd alloy the computed force constants are given in table IV. The average mass used in the calculations for the alloy is obtained from the relation

M( Fe-Pd ) 4 (12X) M( Fe )1XM( Pd ) .

(11)

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. involving a single parameter [9], where a three-body potential is used to deduce the force-constant matrix. For a fcc system, the elements of the diagonal and off-diagonal matrix may be given, after solving the usual secular determinant, as

. / ´ Daam4 4 g[ 4 2 2 C2 i2 Ci(Cj1 Ck) ] , Dm ab4 4 g[Ci(Cj1 Ck) 22] , (12)

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. For TABLEIV. – Computed radial (aj) and tangential (bj) force constants.

Serial no. aj( 1023Nm21) bj( 1023Nm21) Fe Pd Fe-28% Pd Fe Pd Fe-28% Pd 1 49998 56354 51777.6 2260.417 2221.37 2249.483 2 21324.91 21878.20 21479.83 146.893 161.800 151.067 3 2174.090 2139.738 2164.471 16.0348 9.07550 14.0862 4 232.4052 213.8906 227.2211 2.62244 0.77063 2.10393 5 27.51840 21.79550 25.91598 0.54960 0.08877 0.42056 6 22.03252 20.28327 21.54273 0.13662 0.01276 0.10941 7 20.61574 20.05211 20.45792 0.03853 2.17 P 1023 _0.02835 8 20.20238 20.01082 20.14983 0.01198 4.21 P 1024 _8.75P1023

the alloy Fe-28%Pd, the three-body force constant is obtained from the linear relation

g( Fe-Pd ) 4 (12X) g( Fe )1Xg( Pd ) .

(13)

The computed values of the three-body force constants at the lattice constant of the alloy are g420.5855 Nm21 _{for Fe, g421.1319 Nm}21 _{for Pd, and g420.7385 Nm}21

for Fe-28%Pd. Now one can construct the dynamical matrix Dab by using eq. (8) and then solve the secular equation (7) to compute the phonon frequencies along the principal symmetry directions [100], [110] and [111] for the alloy.

3. – Results and discussions

In the present analysis, the interaction system of the fcc Fe-%28Pd alloy is considered to be composed of two-body and three-body parts. By a three-body interaction we mean an extra interaction energy owing to the presence of a third particle. This type of interaction may occur through the deformation of the electron shells [22]. The three-body (2) and two-body (1) model potentials are used, as an application, to investigate the dynamical behaviors of the binary type-II alloys, where the end members have different structures. In the case of the two-body interaction, we have considered the couplings extending to the eighth neighbor of the fcc structure. The parameters (a , r0, D) defining the model potential f2(rj) for pure Fe and Pd are evaluated for fcc structure at the equilibrium lattice constant of the alloy, by the knowledge of the equilibrium pair energies and the equilibrium force constants of the elements. The three-body parameter C is evaluated from the knowledge of the total

Fig. 1. – Phonon dispersion curves at room temperature for Fe-28%Pd. The symbols m,),j represent the experimental values [13]. 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.

cohesive energies (total interaction energies) of the elements. Thus, on the one hand, we determine the ab initio radial and tangential force constants of Fe and Pd for the fcc structure at the lattice constant of the alloy by using the model potential (1) and, on the other, we reasonably account for the long-range character of interatomic forces by considering the interaction system extending up to eighth sets of nearest neighbors. The three-body force constants for Fe and Pd are also calculated for the fcc structure at the lattice constant of the alloy, by using the three-body potential (2). In order to study the phonon dispersion relations of the alloy Fe-28%Pd, we calculate the radial, tangential and three-body force constants of the alloy, using the concentration averages of the force constants of the constituent metals.

First, the phonon dispersion curves of the alloy are only computed according to the two-body central interactions and the results are shown by dashed curves in fig. 1. Next the calculation is repeated for the alloy, by including the contribution of three-body forces to the dynamical matrix (8) and the computed dispersion curves are shown by solid curves in fig. 1. Furthermore, the experimental data measured by Sato

et al. [13] for the alloy is also shown by the symbols l, 1, i in fig. 1 for comparison.

Figure 1 shows that the experimental and theoretical values are in good agreement when the three-body forces are incorporated in the alloy. According to this result, the input data being independent of phonon frequencies and elastic constants enhance the reliability and credibility of the present analysis.

Recently, Garg et al. [23] have studied the dynamical behavior of the alloy. They use deLaunay’s angular force model [24] for the calculation of the force constants of the alloy, assuming the two-body interatomic forces effective up to second-nearest neighbors. Thus, these workers essentially use the short-range forces to study the lattice dynamics of the alloy. However, the interatomic forces in metallic systems are known to possess the long-range character. Furthermore, it is to be pointed out that the force constants of the alloy in this work [23] are evaluated by fitting to the experimental phonon frequencies and the elastic constants of it. In the present work, we account the long-range character of the radial and tangential force constants determined from the model potential (1), by considering the interaction system extending up to eighth sets of the nearest neighbors. Furthermore, the input data in the mean-crystal model expressed in sect. 2 is independent of the phonon frequencies and elastic constants of the alloy and metals.

Consequently, the present results show that the proposed two- and three-body potentials are sufficient to reproduce the phonon data and the scheme described in sect. 2 works well for the fcc binary type-II alloys.

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