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Phonon dispersions and elastic constants of disordered Pd–Ni alloys
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Physica B 355 (2005) 382–391 www.elsevier.com/locate/physb
Phonon dispersions and elastic constants of disordered Pd–Ni alloys S. O¨zdemir Karta,,1, M. Tomaka, T. C - ag˘ınb a Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Department of Chemical Engineering, Texas A&M University, College Station, TX 77845-3122, USA
b
Received 28 October 2003; received in revised form 24 October 2004; accepted 8 November 2004
Abstract Phonon frequencies of Pd–Ni alloys are calculated by molecular dynamics (MD) simulation. Lattice dynamical properties computed from Sutton–Chen (SC) and quantum Sutton–Chen (Q-SC) potentials as a function of temperature are compared with each other. We present all interatomic force constants up to the 8th nearest-neighbor shell obtained by using the calculated potential. Elastic constants evaluated by two methods are consistent with each other. The transferability of the potential is also tested. The results are in good agreement with experimental data and other calculations. r 2004 Elsevier B.V. All rights reserved. PACS: 61.43.Dq; 62.20.Dc; 63.20.Dj Keywords: Molecular dynamics; Sutton–Chen potential; Transition metals; Disordered alloys; Phonon dispersion relations; Elastic constants
1. Introduction The study of the lattice dynamical properties of metals and their alloys remains as one of the most interesting topics, both theoretically and experiCorresponding author. Tel.: +90 3122104332; fax: +90 3122101281. E-mail address: [email protected] (S.O¨. Kart). 1 Permanent address: Department of Physics, Pamukkale University, Denizli, Turkey.
mentally. The easiest test of dynamical properties is the calculation of phonon dispersion curves, for which experimental measurements are available. In particular, the knowledge of the phonon spectrum is an essential input to the calculation of heat capacities, thermal expansion coefficients, electron–phonon interactions, etc. Calculations of the phonon spectrum for FCC transition metals have been carried out in the past mostly using pair potentials [1–4]. Recently, in a work by Singh [5] structural phase transformations
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.11.066
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of Cu, Pd and Au using transition metal pair potentials (TMPP) [6] were studied and it was stated that one should consider many-body potential instead of pair potential to reproduce all properties of the transition metal including bulk modulus. Brovman and Kagan [7] realized the importance of many-body forces in the lattice dynamics of metals and showed that the terms higher than second order in the perturbation expansion are necessary for the equality of the values of compressibility calculated by long waves method and the method of homogeneous deformations. This indicates that many-body potentials are essential to account for long-wavelength phonons. Pair potentials alone are very successful but insufficient to describe metallic bonding which is due to the sharing of the electrons in the system and neglect the physics of metallic bonding. For example, pair potential gives the Cauchy relation of the elastic constants, C 12 ¼ C 44 ; which does not hold for metals. We know that, in metals and their alloys, the many-body density term plays a significant role in the interactions. The model selected for transition metals should include manybody interactions as well as pairwise interactions. Recently, first-principle calculations on the phonon dispersion relations of some FCC transition metals and alloys have been performed successfully [8–12]. This method supplies accurate information about the atomic interactions, but it needs considerable computer power and time to obtain the results. Computer limitations allow this method to simulate systems with only a few hundred atoms. On the other hand, there are empirical and more practical approaches that can afford to investigate many systems and trends in physical properties [13–16]. These potentials provide sufficiently accurate and quick description in metallic systems. For these reasons, the empirical many-body potentials are useful and has recently been applied to phonon properties of some FCC transition metals [17–19] and those of Cu–Au alloy [20,21]. Pseudopotential calculations are still being carried out, aiming to describe lattice mechanical properties of FCC transition metals [22,23]. The choice of Pd–Ni system for the present study is motivated by its technological importance. It is used in hydrogen sensors. This alloy has also
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glass forming properties at the eutectic region around 45% of Pd. At this concentration, Pd–Ni system forms a random disordered alloy with highconcentration mass and it is convenient for an experiment by inelastic neutron scattering [24]. Due to available experimental data, it has been investigated by several workers [25–27]. They used the transition metal model potential (TMMP) of Animalu [28] to obtain phonon dispersion curves of Pd0.45Ni0.55. However, in general, the transverse branches are not in good agreement with the experimental data as computed by using the TMMP. This discrepancy is due to the TMMP of Animalu which treats the pseudopotential in second-order perturbation theory and accounts for the two-body forces only while neglecting the contribution of the higher-order terms. Recently, a many-body potential has been introduced by Sutton and Chen [16] within the context of tight binding approach. This method describes the FCC metals well. Because of computational efficiency and fairly long-range properties, it has been used in many studies and successfully applied to a range of problems [29–35]. Recently, this potential has been applied to study thermal and mechanical properties of some FCC transition metals [36], Pt–Rh alloys [37] and Pd–Ag alloys [38]. So far, the phonon properties of Pd0.45Ni0.55 computed by two or three-body interaction potentials have been published. Hence, our aim is to simulate Pd–Ni system by using many-body potentials and to see their effects on the lattice dynamical properties. In this study, we have performed molecular dynamics (MD) simulations using the Sutton–Chen (SC) potential and its new potential parameter set, quantum Sutton–Chen potential (Q-SC) developed by C - ag˘ın and coworkers [39]. We are particularly interested in the concentration and the temperature dependence on phonon dispersion relation and elastic constants to show the validity of potential energy function. We also intend to test the transferability of the potential from pure elemental form to the alloy without further empirical fitting to properties of Pd–Ni alloys. To the best of our knowledge this work presents the phonon dispersion curves of alloys focusing on Pd–Ni by using the SC potentials for the first time. The elastic constants
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are determined both from the long-wavelength limit of the phonon dispersion and from static deformation of unit cell by taking the second derivative of the total energy with respect to the deformations. The performance of the Q-SC potential on these two properties are compared with the SC and the other potential results. This paper is organized as follows: the manybody potential model and the method of calculation are described in Section 2. The methods we follow to investigate the phonon dispersion relation is presented in the same section. In Section 3, we show and discuss the numerical results. Whenever possible, these results are compared with experimental values and other calculations. Finally, conclusions are given in Section 4.
2. Methods 2.1. Sutton– Chen potential and computational procedure The SC interaction potential, long-range Finnis– Sinclair potential for FCC transition metals, consists of a pairwise repulsive part and manybody attractive part. The total potential energy of the metal alloys has the following form: X U tot ¼ Ui i
" # X X1 1=2 ij V ðrij Þ ci ij ðri Þ ¼ ; 2 i jai
ð1Þ
where V ðrij Þ is a pairwise potential describing longrange interaction with a van der Walls tail between the i and j atomic cores nij aij V ðrij Þ ¼ ; (2) rij and ri is a local energy density responsible for cohesive interaction at short range associated with atom i, given as X X aij mij ri ¼ fðrij Þ ¼ : (3) rij jai jai In Eqs. (1)–(3), a is a parameter with the dimensions of length, c is a positive dimensionless
parameter scaling the cohesive term relative to repulsive term, sets the overall energy scale, and n, m are material parameters, such that n4m: The combination rules to extend the SC model to alloys are in the following forms [40]: pffiffiffiffiffiffiffi (4) ij ¼ i j ; mij ¼
mi þ mj ; 2
(5)
nij ¼
ni þ nj ; 2
(6)
ai þ aj : (7) 2 These SC parameters are obtained by fitting to the 0 K properties based on the experimental lattice parameter, cohesive energy and bulk modulus. Recently, C - ag˘ın and co-workers [39] modified the SC potential by including quantum corrections (e.g., zero-point energy) to improve the results of the potential for elevated temperatures. The quantum Sutton–Chen (Q-SC) parameters are optimized to fit to additional experimental properties, such as phonon frequencies at the X point (at room temperature), vacancy formation energy and surface energies. In Table 1, the values of the Q-SC and SC parameters for Pd and Ni are listed. C - ag˘ın and co-workers have used these potentials for various applications ranging from alloys, glass formations, crystallization, surface science problems, clusters, nanowires, and single crystal plasticity of pure metals to transport properties of FCC transition metals [41–45]. We have carried out MD simulations whose algorithms are based on extended Hamiltonian aij ¼
Table 1 Quantum Sutton–Chen (Q-SC) [39] and Sutton–Chen (SC) [40] potential parameters Metal
Model
n
m
(eV)
c
a (A1)
Pd
Q-SC SC
12 12
6 7
3.2864E-3 4.1260E-3
148.205 108.526
3.8813 3.8900
Ni
Q-SC SC
10 9
5 6
7.3767E-3 1.5714E-2
84.745 39.756
3.5157 3.5200
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formalism [46–49]. SC and Q-SC potential parameters are used to describe the interactions between the atoms. We adopt the random binary FCC metal alloy method developed by RafiiTabar and Sutton [40]; two types of atoms occupies the sites completely randomly. The system is made up of a cubic box with 864 atoms. At the beginning of the simulation, atoms are randomly arranged on a FCC lattice subject to periodic boundary conditions in three dimensions. Newton’s equation of motion is solved by using 5th order Gear predictor–corrector algorithm with a time step of Dt ¼ 0:002 ps: Three successive simulations are performed for heating Pd–Ni alloys (Pdx Ni1x ; where x ¼ 0; 0:1; 0:2; . . . ; 1; and x ¼ 0:45). First, HPN (constant-enthalpy and constant-pressure) MD simulation is carried out to heat the system from 0.1 K to target temperature with increments of 200 K. 2000 time steps are carried out for equilibrium, at each temperature. Then 20,000 additional steps in TPN (constanttemperature and constant-pressure) dynamics are taken to obtain some statistical production; such as volume, density and energy of the system. Finally, 50,000 steps of EVN (microcanonical) dynamics are performed by using the resulting zero strain average matrix /h0 S to obtain pressure dependent properties of the system, such as elastic constant and phonon dispersion relation. Parrinello–Rahman piston mass parameter W and Nose´-Hoover thermostat mass Q are chosen 400 and 100, respectively. The cutoff distance for the interactions between the atoms is taken as two lattice parameters where the forces are negligibly small. An additional distance of half a lattice parameter is also added to this range to consider the temperature effect. 2.2. Phonon dispersion relations The phonon frequencies can be obtained by solving the secular equation [50,51] 2
det jDab;ln ðqÞ w dab dln j ¼ 0;
ð8Þ
where w is the angular frequency of the vibration, q is the wave vector of the crystal vibration, dab is the Kronecker symbol and Dab;ln is an element of
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the so-called dynamical matrix [51] Dab;ln ðqÞ ¼
1
X
ðml mn Þ1=2
l0
Fab
l
l0
l
n
!
0
expðiq:½rðn; l Þ rðl; lÞ Þ;
ð9Þ
where Fab is the force constant, ml is the mass of the lth atom, l refers to the reference unit cell, l 0 denotes the neighbor unit cell. The force constant can be computed by taking first derivative of the many-body force given in the previous works [36,37] Fab ði; jÞ
"
¼
þ
ðV 00ij
X
c 3=2 4ri
c
"
1=2
þ
! f0ik Drika
c
f0 Drija 3=2 ij 4rj c 1=2
2rj
þ
V 0ij
dab r2ij
#
f0ij Drijb
kai
ðf00ij
2ri þ
V 0ij ÞDrija Drijb
"
ðf00ij
f0ij ÞDrija Drijb X
þ
f0ij
dab r2ij
f0ij
dab r2ij
! f0jk Drjkb
kaj
f0ij ÞDrija Drijb
þ
" # 1 X c ðf0 Drika f0jk Drjkb Þ ; þ 4 kai;j r3=2 ik k and Fab ði; iÞ ¼
X
Fab ði; jÞ;
#
#
ð10Þ
ð11Þ
jai
here 0 is rij q=qrij ; 00 refers to r2ij q2 =qr2ij , i is the index for the lth atom of the lth lattice cell ðl; lÞ and j is the index for the nth atom of l 0 th lattice cell ðn; l 0 Þ: Drij is expressed by the position of the atoms; rai raj Drija ¼ ; ð12Þ r2ij rai is the position of the ith particle in the direction a: We have computed the interatomic force constants to obtain the phonon dispersion curves. The trajectories of the EVN dynamics which are averaged are used to calculate the force constants.
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The system size for the alloy, 864 atoms, is insufficient for obeying the symmetry rules for the force constants. Therefore we construct the dynamical matrix composed of 12 12 elements by using 1372 atoms (7 7 7 cubic system) for the alloy system for the high symmetry directions in the first Brillouin zone.
3. Results and discussion The force constants of Pd–Ni system are calculated by using the expression given in Eq. (10). We study the effective interaction range in order to demonstrate the sensitivity of the force constants on the cutoff. The force constants decay rapidly with distance and they are negligible beyond the 8th nearest-neighbor shell. Table 2 shows the non-vanishing elements of the force constants for Pd, Ni and Pd0.45Ni0.55. Comparison with the force constants calculated to the neutron measurements [24] reveals good agreement. Phonon dispersion curves for Pd at 120 K, Ni at 300 K and Pd0.45Ni0.55 at 300 K are presented in Figs. 1–3, respectively. The calculations are performed by using both the SC parameters (dashed lines) and the Q-SC parameters (solid lines). The plots are shown along three principal symmetry directions, together with the experimental data [24,52,53]. While Q-SC potential improves the description of phonons for Ni and Pd0.45Ni0.55,
it does not show any improvement on the phonon frequencies for Pd. Generally Q-SC calculations are in better agreement with experimental values. However, we may have to underline the fact that although SC potential has been fitted to only three experimental quantities, it still describes phonon properties. The phonon spectra of Pd0.45Ni0.55 obtained from Q-SC is consistent with the experiment as shown in Fig. 3. This encourages us to study the phonon dispersion relations at other concentrations of the Pd–Ni system, which are not
Fig. 1. Phonon dispersion curves of Pd along symmetry directions. The solid curves represent the present calculations using Q-SC parameters. The dashed curves show the SC calculations. The points are the experimental data by Miller and Brockhouse [52] at 120 K.
Table 2 Force constants of Pd, Ni and Pd0.45Ni0.55 in the units of dyn/cm up to the 8th nearest-neighbor (NN) shell by using Q-SC potential parameters NN
Comp.
Force constants Pd
1 2 3 4 5 6 7 8
xx xx xx xz xx xx xz xx xx yz xx
zz yy yy
xy
zz yy
xy zz
yy yy xz yy
zz xy
yz
16980 528 265 171 200 44 21 0.5 10 3 6
Pd0.45Ni0.55
Ni 1532 262 34
18129
8 7
202 44
10 0.6 7 0.7
4 3
139
17447 783 63 84 113 176 11 2.63 8 2 4
1914 85 37
19111
112 4
105 18
6 0.3 5 0.05
24
0.3 2
18095 666 218 310 192 34 16 7 8 3 10
2776 245 30
25300
12 4
194 5
14 3 2 1
112
0.7 5
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Fig. 2. Phonon dispersion curves of Ni along symmetry directions. The points are the experimental data by Birgeneau et al. [53] at 300 K. The other symbols are the same as that of Fig. 1.
Fig. 3. Phonon dispersion curves of Pd0.45Ni0.55 along symmetry directions. The points are the experimental data by Kamitakahara and Brockhouse [24] at 300 K. The other symbols are the same as that of Fig. 1.
available in the literature. The results for the transverse (T) and longitudinal (L) phonon frequencies for Pd–Ni alloys at X and L points at the Brillouin zone are given in Fig. 4. As the concentration of Pd increases in Ni, transverse frequencies decrease linearly (Figs. 4a and c). On the other hand, longitudinal frequencies show polynomial behavior as given in Figs. 4b and d. Calculation of the phonon dispersion curves as a
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function of temperature can be useful to identify unstable phonon modes. Hence, we are interested in variation of frequencies with temperature. The frequencies at various temperatures for Pd and Ni at the Brillouin zone are listed in Table 3. As expected, the frequency of phonons decreases with temperature. Q-SC simulation results at 300 K are compared with experimental data [54] in the same table. Also included in Table 3 are the other simulation results. The Q-SC results are more compatible with the experimental values than the results of previous works using tight binding second moment approximation [17] and embedded atom models [55,56] except for the transverse modes of Ni. The best agreement with the experiment is the longitudinal frequency at X point (nXL ) for Ni, as expected, due to fitting of Q-SC parameters to experimental phonon frequencies. As a further test for the derived force constants, the elastic constants via the method of long waves which relates the elastic constants calculated by the slopes of the dispersion curves as k ! 0; are calculated. However, this is a very delicate procedure since small errors in long-range couplings strongly influence the result. Alternatively, the elastic constants can be derived via homogeneous deformation of unit cell from the second derivative of the total energy with respect to the deformation. The second direct approach is provided by using the statistical fluctuation formulae given in Ref. [57]. The elastic properties of bulk Pd–Ni alloys are calculated by using both SC and Q-SC parameters over 50,000 steps in the microcanonical ensemble (EVN), hence resulting in the adiabatic elastic constants. Table 4 compares our results for the elastic constants at 300 K as obtained with two methods to experimental values [58]. A reasonable consistency of both sets of elastic constants for two potentials is achieved, except for elastic constants of C 44 for Ni (the discrepancy is about 19%). As shown in Table 4, the values for Ni derived from the long waves method are in more reasonable agreement with experiment than those based on homogeneous deformation method. This is due to the consistency of phonon dispersion curves with the experiment.
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Fig. 4. The phonon frequencies for (a) the transverse branches at the X point (nXT ), (b) the longitudinal branch at the X point (nXL ), (c) the transverse branches at the L point (nLT ), (d) the longitudinal branch at the L point (nLL ) at the Brillouin zone as a function of concentration of Pd in Ni at 300 K. The points are simulation results. Our data are fitted to the functions given in the figures.
Table 3 Comparison of the calculated and experimental transverse (T) and longitudinal (L) phonon frequencies at the Brillouin zone boundaries X and L for Pd and Ni Metal
Model
T (K)
nXT (THz)
nXL (THz)
nLT (THz)
nLL (THz)
Pd
Q-SC
300 500 700 900 300 300 300
4.25 4.08 3.93 2.64 4.56 4.02 5.91
6.19 5.93 5.68 3.89 6.70 5.77 4.09
2.83 2.73 2.60 2.49 3.21 2.67 2.66
6.17 5.90 5.62 5.30 6.86 5.72 5.88
300 500 700 900 300 300 300
5.68 5.52 5.33 5.11 6.17 6.78 6.78
8.48 8.20 7.90 7.58 8.55 9.88 9.90
3.69 3.58 3.46 3.36 4.24 4.49 4.52
8.50 8.21 7.90 7.55 8.88 9.80 9.77
Exp tbsma eam Ni
Q-SC
Exp tbsma eam
The first four rows for Pd and Ni are the calculations using QSC at various temperatures. Our results at 300 K are compared with the experimental (Exp) [54] and the other two potential results: tight binding second moment approximation (tbsma) [17] and embedded atom model (eam) [55,56].
The results obtained from the homogeneous deformation method for pure metals at 0 K and alloys at 300 K are given in Table 5, along with comparison with experimental data, whenever available, and the results of previous works using different potential models. As shown, the potential used by Cleri and Rosato [17] for the elastic constants is yielding better results. The elastic constants and bulk modulus both from Q-SC and SC calculations agree with the available experiment and the other calculations except for elastic constant of C 44 : The only metal alloy studied here for which the first-principle calculations are available is Pd. While the accuracy of the Q-SC elastic constant of C 11 for Pd, showing a deviation of 8% from experiment, is comparable to that of the embedded atom model (eam) [59] and firstprinciple calculations (LAPW) [60], SC elastic constant of C 12 is in good agreement with experiment. The experimental data on the elastic constants for Pd–Ni alloys are not available for comparison. Our results for Pd0.45Ni0.55 are smaller than the values calculated by Upadhyaya et al. [25] using the TMMP of Animalu [28] except
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Table 4 Elastic constants of Pd and Ni at 300 K in the units of GPa Metal
Els. Cons.
Exp
Q-SC
SC
H
L
DHL ð%Þ
H
L
DHL ð%Þ
Pd
C 11 C 12 C 44
227.1 176.0 71.7
201.89 140.15 81.44
186.71 125.67 79.78
7.1 10.3 2.0
229.48 164.45 82.72
214.21 149.74 81.29
6.7 8.9 1.7
Ni
C 11 C 12 C 44
250.8 150.0 123.5
214.57 152.95 89.04
237.42 164.53 105.93
10.7 7.6 19.0
213.69 166.10 69.91
237.55 180.34 83.36
11.2 8.6 19.2
The calculated results (with Q-SC and SC) are obtained by homogeneous deformations ðHÞ and long waves ðLÞ methods. Experimental values are taken from Ref. [58]. Table 5 Comparison of calculated and experimental (Exp) [58] values for elastic constants (C ij ) and bulk modulus ðBÞ at 0 K for pure metals and at 300 K for Pd–Ni alloys in the units of GPa Metal
Model
C 11
C 12
C 44
B
Pd
Exp Q-SC SC eam tbsma LAPW
234.12 216.00 248.20 218.00 232.00 218.00
176.14 150.25 175.90 184.00 178.00 172.00
71.17 91.66 93.29 65.00 73.00 74.00
195.00 172.17 200.00 195.00 196.00
Pd0.8Ni0.2 Pd0.6Ni0.4
Q-SC Q-SC
198.83 197.63
145.69 150.64
79.37 77.83
163.02 165.75
Pd0.45Ni0.55
Q-SC SC TMMP
199.13 214.97 224.50
153.29 174.33 154.70
77.69 69.34 94.50
168.57 187.88
Pd0.4Ni0.6 Pd0.2Ni0.8
Q-SC Q-SC
199.28 204.81
152.54 154.37
78.20 82.26
168.00 170.78
Ni
Exp Q-SC SC eam tbsma
261.20 219.59 226.61 233.00 257.00
150.80 165.34 178.77 154.00 155.00
131.70 99.57 79.24 128.00 136.00
188.00 183.62 194.73 180.40 189.00
Our results (Q-SC and SC) for Pd and Ni are compared with the other potential models: embedded atom method (eam) [59], tight binding second moment approximation (tbsma) [17], firstprinciples full potential (LAPW) [60], where available. A comparison for Pd0.45Ni0.55 is made between our results of QSC and SC calculations and transition metal model potential (TMMP) [25].
for C 12 calculated from SC calculations. We are also interested in temperature dependence of elastic constants in order to see whether there is
an improvement of Q-SC on the results. As the temperature increases, the Q-SC parameters improve the elastic constants slightly, as shown in Table 6. Thermal softening of alloy occurs with increasing temperature.
4. Conclusion We have presented MD simulations of the lattice dynamics of disordered Pd–Ni metal alloys by using Q-SC and SC potential parameters in this study. The transferability of the potential is an important conclusion which can be made from this work. Although the potential parameters were fitted to solid experimental properties of the pure system, the Q-SC model describes the lattice properties of Pd–Ni alloys. The contributions up to the 8th nearest-neighbor shell to force constants are found sufficient to achieve convergence for the phonon dispersion calculations. Taking into account the other Nbody potentials (see Table 3), Q-SC potential seems to give quite an accurate description of phonon properties. Even though Q-SC potential is fitted to phonon frequencies at only X point, the overall structure of dispersion curves are well reproduced. This potential produces better results for the transverse phonon frequencies than those obtained by pair potentials [25–27]. We may also concluded that SC potential agrees with experimental results despite fitting not to phonon frequencies. This work also presents the phonon
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Table 6 Comparison of calculated (Q-SC and SC) and experimental (Exp) [58] values for elastic constants (C ij ) of Ni in the units of GPa at various temperatures T (K)
0 300 500 700
Exp
Q-SC
SC
C 11
C 12
C 44
C 11
C 12
C 44
C 11
C 12
C 44
261.20 250.80 239.60 226.10
150.80 150.00 149.00 147.30
131.70 123.50 115.90 107.90
216.33 214.57 203.87 193.34
165.337 152.95 147.53 141.75
99.39 89.04 82.79 76.35
251.736 213.69 201.73 188.20
191.12 166.10 158.84 151.21
89.44 69.91 63.58 56.65
dispersion curves of alloys by using the SC potentials for the first time. Elastic constants for Pd–Ni alloys, except for Pd and Ni, are also calculated by using many-body potential for the first time by this study. Two methods have been used; homogeneous deformation and long waves. Our results from both methods are comparable for each potential parameters. Hence the consistency condition of matching the elastic constants calculated by two methods are obtained. We have seen that Q-SC potential improves the elastic constants slightly as the temperature increases. The experimental data on the elastic constants for Pd–Ni alloys except for Pd and Ni are not available for comparison. These data may encourage the experimentalists to verify our results. In summary, our work shows that Q-SC potential may be used to predict the lattice dynamics of FCC transition metals and their alloys with sufficient accuracy and to provide a detailed understanding of phonon properties. This study may give a helpful guide to researchers to improve the potential parameters for real systems. The simulation results may be further improved by fitting to experimental or first-principle calculation of solid properties of the alloy.
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Physica B 355 (2005) 382–391 www.elsevier.com/locate/physb
Phonon dispersions and elastic constants of disordered Pd–Ni alloys S. O¨zdemir Karta,,1, M. Tomaka, T. C - ag˘ınb a Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Department of Chemical Engineering, Texas A&M University, College Station, TX 77845-3122, USA
b
Received 28 October 2003; received in revised form 24 October 2004; accepted 8 November 2004
Abstract Phonon frequencies of Pd–Ni alloys are calculated by molecular dynamics (MD) simulation. Lattice dynamical properties computed from Sutton–Chen (SC) and quantum Sutton–Chen (Q-SC) potentials as a function of temperature are compared with each other. We present all interatomic force constants up to the 8th nearest-neighbor shell obtained by using the calculated potential. Elastic constants evaluated by two methods are consistent with each other. The transferability of the potential is also tested. The results are in good agreement with experimental data and other calculations. r 2004 Elsevier B.V. All rights reserved. PACS: 61.43.Dq; 62.20.Dc; 63.20.Dj Keywords: Molecular dynamics; Sutton–Chen potential; Transition metals; Disordered alloys; Phonon dispersion relations; Elastic constants
1. Introduction The study of the lattice dynamical properties of metals and their alloys remains as one of the most interesting topics, both theoretically and experiCorresponding author. Tel.: +90 3122104332; fax: +90 3122101281. E-mail address: [email protected] (S.O¨. Kart). 1 Permanent address: Department of Physics, Pamukkale University, Denizli, Turkey.
mentally. The easiest test of dynamical properties is the calculation of phonon dispersion curves, for which experimental measurements are available. In particular, the knowledge of the phonon spectrum is an essential input to the calculation of heat capacities, thermal expansion coefficients, electron–phonon interactions, etc. Calculations of the phonon spectrum for FCC transition metals have been carried out in the past mostly using pair potentials [1–4]. Recently, in a work by Singh [5] structural phase transformations
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.11.066
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of Cu, Pd and Au using transition metal pair potentials (TMPP) [6] were studied and it was stated that one should consider many-body potential instead of pair potential to reproduce all properties of the transition metal including bulk modulus. Brovman and Kagan [7] realized the importance of many-body forces in the lattice dynamics of metals and showed that the terms higher than second order in the perturbation expansion are necessary for the equality of the values of compressibility calculated by long waves method and the method of homogeneous deformations. This indicates that many-body potentials are essential to account for long-wavelength phonons. Pair potentials alone are very successful but insufficient to describe metallic bonding which is due to the sharing of the electrons in the system and neglect the physics of metallic bonding. For example, pair potential gives the Cauchy relation of the elastic constants, C 12 ¼ C 44 ; which does not hold for metals. We know that, in metals and their alloys, the many-body density term plays a significant role in the interactions. The model selected for transition metals should include manybody interactions as well as pairwise interactions. Recently, first-principle calculations on the phonon dispersion relations of some FCC transition metals and alloys have been performed successfully [8–12]. This method supplies accurate information about the atomic interactions, but it needs considerable computer power and time to obtain the results. Computer limitations allow this method to simulate systems with only a few hundred atoms. On the other hand, there are empirical and more practical approaches that can afford to investigate many systems and trends in physical properties [13–16]. These potentials provide sufficiently accurate and quick description in metallic systems. For these reasons, the empirical many-body potentials are useful and has recently been applied to phonon properties of some FCC transition metals [17–19] and those of Cu–Au alloy [20,21]. Pseudopotential calculations are still being carried out, aiming to describe lattice mechanical properties of FCC transition metals [22,23]. The choice of Pd–Ni system for the present study is motivated by its technological importance. It is used in hydrogen sensors. This alloy has also
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glass forming properties at the eutectic region around 45% of Pd. At this concentration, Pd–Ni system forms a random disordered alloy with highconcentration mass and it is convenient for an experiment by inelastic neutron scattering [24]. Due to available experimental data, it has been investigated by several workers [25–27]. They used the transition metal model potential (TMMP) of Animalu [28] to obtain phonon dispersion curves of Pd0.45Ni0.55. However, in general, the transverse branches are not in good agreement with the experimental data as computed by using the TMMP. This discrepancy is due to the TMMP of Animalu which treats the pseudopotential in second-order perturbation theory and accounts for the two-body forces only while neglecting the contribution of the higher-order terms. Recently, a many-body potential has been introduced by Sutton and Chen [16] within the context of tight binding approach. This method describes the FCC metals well. Because of computational efficiency and fairly long-range properties, it has been used in many studies and successfully applied to a range of problems [29–35]. Recently, this potential has been applied to study thermal and mechanical properties of some FCC transition metals [36], Pt–Rh alloys [37] and Pd–Ag alloys [38]. So far, the phonon properties of Pd0.45Ni0.55 computed by two or three-body interaction potentials have been published. Hence, our aim is to simulate Pd–Ni system by using many-body potentials and to see their effects on the lattice dynamical properties. In this study, we have performed molecular dynamics (MD) simulations using the Sutton–Chen (SC) potential and its new potential parameter set, quantum Sutton–Chen potential (Q-SC) developed by C - ag˘ın and coworkers [39]. We are particularly interested in the concentration and the temperature dependence on phonon dispersion relation and elastic constants to show the validity of potential energy function. We also intend to test the transferability of the potential from pure elemental form to the alloy without further empirical fitting to properties of Pd–Ni alloys. To the best of our knowledge this work presents the phonon dispersion curves of alloys focusing on Pd–Ni by using the SC potentials for the first time. The elastic constants
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are determined both from the long-wavelength limit of the phonon dispersion and from static deformation of unit cell by taking the second derivative of the total energy with respect to the deformations. The performance of the Q-SC potential on these two properties are compared with the SC and the other potential results. This paper is organized as follows: the manybody potential model and the method of calculation are described in Section 2. The methods we follow to investigate the phonon dispersion relation is presented in the same section. In Section 3, we show and discuss the numerical results. Whenever possible, these results are compared with experimental values and other calculations. Finally, conclusions are given in Section 4.
2. Methods 2.1. Sutton– Chen potential and computational procedure The SC interaction potential, long-range Finnis– Sinclair potential for FCC transition metals, consists of a pairwise repulsive part and manybody attractive part. The total potential energy of the metal alloys has the following form: X U tot ¼ Ui i
" # X X1 1=2 ij V ðrij Þ ci ij ðri Þ ¼ ; 2 i jai
ð1Þ
where V ðrij Þ is a pairwise potential describing longrange interaction with a van der Walls tail between the i and j atomic cores nij aij V ðrij Þ ¼ ; (2) rij and ri is a local energy density responsible for cohesive interaction at short range associated with atom i, given as X X aij mij ri ¼ fðrij Þ ¼ : (3) rij jai jai In Eqs. (1)–(3), a is a parameter with the dimensions of length, c is a positive dimensionless
parameter scaling the cohesive term relative to repulsive term, sets the overall energy scale, and n, m are material parameters, such that n4m: The combination rules to extend the SC model to alloys are in the following forms [40]: pffiffiffiffiffiffiffi (4) ij ¼ i j ; mij ¼
mi þ mj ; 2
(5)
nij ¼
ni þ nj ; 2
(6)
ai þ aj : (7) 2 These SC parameters are obtained by fitting to the 0 K properties based on the experimental lattice parameter, cohesive energy and bulk modulus. Recently, C - ag˘ın and co-workers [39] modified the SC potential by including quantum corrections (e.g., zero-point energy) to improve the results of the potential for elevated temperatures. The quantum Sutton–Chen (Q-SC) parameters are optimized to fit to additional experimental properties, such as phonon frequencies at the X point (at room temperature), vacancy formation energy and surface energies. In Table 1, the values of the Q-SC and SC parameters for Pd and Ni are listed. C - ag˘ın and co-workers have used these potentials for various applications ranging from alloys, glass formations, crystallization, surface science problems, clusters, nanowires, and single crystal plasticity of pure metals to transport properties of FCC transition metals [41–45]. We have carried out MD simulations whose algorithms are based on extended Hamiltonian aij ¼
Table 1 Quantum Sutton–Chen (Q-SC) [39] and Sutton–Chen (SC) [40] potential parameters Metal
Model
n
m
(eV)
c
a (A1)
Pd
Q-SC SC
12 12
6 7
3.2864E-3 4.1260E-3
148.205 108.526
3.8813 3.8900
Ni
Q-SC SC
10 9
5 6
7.3767E-3 1.5714E-2
84.745 39.756
3.5157 3.5200
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formalism [46–49]. SC and Q-SC potential parameters are used to describe the interactions between the atoms. We adopt the random binary FCC metal alloy method developed by RafiiTabar and Sutton [40]; two types of atoms occupies the sites completely randomly. The system is made up of a cubic box with 864 atoms. At the beginning of the simulation, atoms are randomly arranged on a FCC lattice subject to periodic boundary conditions in three dimensions. Newton’s equation of motion is solved by using 5th order Gear predictor–corrector algorithm with a time step of Dt ¼ 0:002 ps: Three successive simulations are performed for heating Pd–Ni alloys (Pdx Ni1x ; where x ¼ 0; 0:1; 0:2; . . . ; 1; and x ¼ 0:45). First, HPN (constant-enthalpy and constant-pressure) MD simulation is carried out to heat the system from 0.1 K to target temperature with increments of 200 K. 2000 time steps are carried out for equilibrium, at each temperature. Then 20,000 additional steps in TPN (constanttemperature and constant-pressure) dynamics are taken to obtain some statistical production; such as volume, density and energy of the system. Finally, 50,000 steps of EVN (microcanonical) dynamics are performed by using the resulting zero strain average matrix /h0 S to obtain pressure dependent properties of the system, such as elastic constant and phonon dispersion relation. Parrinello–Rahman piston mass parameter W and Nose´-Hoover thermostat mass Q are chosen 400 and 100, respectively. The cutoff distance for the interactions between the atoms is taken as two lattice parameters where the forces are negligibly small. An additional distance of half a lattice parameter is also added to this range to consider the temperature effect. 2.2. Phonon dispersion relations The phonon frequencies can be obtained by solving the secular equation [50,51] 2
det jDab;ln ðqÞ w dab dln j ¼ 0;
ð8Þ
where w is the angular frequency of the vibration, q is the wave vector of the crystal vibration, dab is the Kronecker symbol and Dab;ln is an element of
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the so-called dynamical matrix [51] Dab;ln ðqÞ ¼
1
X
ðml mn Þ1=2
l0
Fab
l
l0
l
n
!
0
expðiq:½rðn; l Þ rðl; lÞ Þ;
ð9Þ
where Fab is the force constant, ml is the mass of the lth atom, l refers to the reference unit cell, l 0 denotes the neighbor unit cell. The force constant can be computed by taking first derivative of the many-body force given in the previous works [36,37] Fab ði; jÞ
"
¼
þ
ðV 00ij
X
c 3=2 4ri
c
"
1=2
þ
! f0ik Drika
c
f0 Drija 3=2 ij 4rj c 1=2
2rj
þ
V 0ij
dab r2ij
#
f0ij Drijb
kai
ðf00ij
2ri þ
V 0ij ÞDrija Drijb
"
ðf00ij
f0ij ÞDrija Drijb X
þ
f0ij
dab r2ij
f0ij
dab r2ij
! f0jk Drjkb
kaj
f0ij ÞDrija Drijb
þ
" # 1 X c ðf0 Drika f0jk Drjkb Þ ; þ 4 kai;j r3=2 ik k and Fab ði; iÞ ¼
X
Fab ði; jÞ;
#
#
ð10Þ
ð11Þ
jai
here 0 is rij q=qrij ; 00 refers to r2ij q2 =qr2ij , i is the index for the lth atom of the lth lattice cell ðl; lÞ and j is the index for the nth atom of l 0 th lattice cell ðn; l 0 Þ: Drij is expressed by the position of the atoms; rai raj Drija ¼ ; ð12Þ r2ij rai is the position of the ith particle in the direction a: We have computed the interatomic force constants to obtain the phonon dispersion curves. The trajectories of the EVN dynamics which are averaged are used to calculate the force constants.
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The system size for the alloy, 864 atoms, is insufficient for obeying the symmetry rules for the force constants. Therefore we construct the dynamical matrix composed of 12 12 elements by using 1372 atoms (7 7 7 cubic system) for the alloy system for the high symmetry directions in the first Brillouin zone.
3. Results and discussion The force constants of Pd–Ni system are calculated by using the expression given in Eq. (10). We study the effective interaction range in order to demonstrate the sensitivity of the force constants on the cutoff. The force constants decay rapidly with distance and they are negligible beyond the 8th nearest-neighbor shell. Table 2 shows the non-vanishing elements of the force constants for Pd, Ni and Pd0.45Ni0.55. Comparison with the force constants calculated to the neutron measurements [24] reveals good agreement. Phonon dispersion curves for Pd at 120 K, Ni at 300 K and Pd0.45Ni0.55 at 300 K are presented in Figs. 1–3, respectively. The calculations are performed by using both the SC parameters (dashed lines) and the Q-SC parameters (solid lines). The plots are shown along three principal symmetry directions, together with the experimental data [24,52,53]. While Q-SC potential improves the description of phonons for Ni and Pd0.45Ni0.55,
it does not show any improvement on the phonon frequencies for Pd. Generally Q-SC calculations are in better agreement with experimental values. However, we may have to underline the fact that although SC potential has been fitted to only three experimental quantities, it still describes phonon properties. The phonon spectra of Pd0.45Ni0.55 obtained from Q-SC is consistent with the experiment as shown in Fig. 3. This encourages us to study the phonon dispersion relations at other concentrations of the Pd–Ni system, which are not
Fig. 1. Phonon dispersion curves of Pd along symmetry directions. The solid curves represent the present calculations using Q-SC parameters. The dashed curves show the SC calculations. The points are the experimental data by Miller and Brockhouse [52] at 120 K.
Table 2 Force constants of Pd, Ni and Pd0.45Ni0.55 in the units of dyn/cm up to the 8th nearest-neighbor (NN) shell by using Q-SC potential parameters NN
Comp.
Force constants Pd
1 2 3 4 5 6 7 8
xx xx xx xz xx xx xz xx xx yz xx
zz yy yy
xy
zz yy
xy zz
yy yy xz yy
zz xy
yz
16980 528 265 171 200 44 21 0.5 10 3 6
Pd0.45Ni0.55
Ni 1532 262 34
18129
8 7
202 44
10 0.6 7 0.7
4 3
139
17447 783 63 84 113 176 11 2.63 8 2 4
1914 85 37
19111
112 4
105 18
6 0.3 5 0.05
24
0.3 2
18095 666 218 310 192 34 16 7 8 3 10
2776 245 30
25300
12 4
194 5
14 3 2 1
112
0.7 5
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Fig. 2. Phonon dispersion curves of Ni along symmetry directions. The points are the experimental data by Birgeneau et al. [53] at 300 K. The other symbols are the same as that of Fig. 1.
Fig. 3. Phonon dispersion curves of Pd0.45Ni0.55 along symmetry directions. The points are the experimental data by Kamitakahara and Brockhouse [24] at 300 K. The other symbols are the same as that of Fig. 1.
available in the literature. The results for the transverse (T) and longitudinal (L) phonon frequencies for Pd–Ni alloys at X and L points at the Brillouin zone are given in Fig. 4. As the concentration of Pd increases in Ni, transverse frequencies decrease linearly (Figs. 4a and c). On the other hand, longitudinal frequencies show polynomial behavior as given in Figs. 4b and d. Calculation of the phonon dispersion curves as a
387
function of temperature can be useful to identify unstable phonon modes. Hence, we are interested in variation of frequencies with temperature. The frequencies at various temperatures for Pd and Ni at the Brillouin zone are listed in Table 3. As expected, the frequency of phonons decreases with temperature. Q-SC simulation results at 300 K are compared with experimental data [54] in the same table. Also included in Table 3 are the other simulation results. The Q-SC results are more compatible with the experimental values than the results of previous works using tight binding second moment approximation [17] and embedded atom models [55,56] except for the transverse modes of Ni. The best agreement with the experiment is the longitudinal frequency at X point (nXL ) for Ni, as expected, due to fitting of Q-SC parameters to experimental phonon frequencies. As a further test for the derived force constants, the elastic constants via the method of long waves which relates the elastic constants calculated by the slopes of the dispersion curves as k ! 0; are calculated. However, this is a very delicate procedure since small errors in long-range couplings strongly influence the result. Alternatively, the elastic constants can be derived via homogeneous deformation of unit cell from the second derivative of the total energy with respect to the deformation. The second direct approach is provided by using the statistical fluctuation formulae given in Ref. [57]. The elastic properties of bulk Pd–Ni alloys are calculated by using both SC and Q-SC parameters over 50,000 steps in the microcanonical ensemble (EVN), hence resulting in the adiabatic elastic constants. Table 4 compares our results for the elastic constants at 300 K as obtained with two methods to experimental values [58]. A reasonable consistency of both sets of elastic constants for two potentials is achieved, except for elastic constants of C 44 for Ni (the discrepancy is about 19%). As shown in Table 4, the values for Ni derived from the long waves method are in more reasonable agreement with experiment than those based on homogeneous deformation method. This is due to the consistency of phonon dispersion curves with the experiment.
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Fig. 4. The phonon frequencies for (a) the transverse branches at the X point (nXT ), (b) the longitudinal branch at the X point (nXL ), (c) the transverse branches at the L point (nLT ), (d) the longitudinal branch at the L point (nLL ) at the Brillouin zone as a function of concentration of Pd in Ni at 300 K. The points are simulation results. Our data are fitted to the functions given in the figures.
Table 3 Comparison of the calculated and experimental transverse (T) and longitudinal (L) phonon frequencies at the Brillouin zone boundaries X and L for Pd and Ni Metal
Model
T (K)
nXT (THz)
nXL (THz)
nLT (THz)
nLL (THz)
Pd
Q-SC
300 500 700 900 300 300 300
4.25 4.08 3.93 2.64 4.56 4.02 5.91
6.19 5.93 5.68 3.89 6.70 5.77 4.09
2.83 2.73 2.60 2.49 3.21 2.67 2.66
6.17 5.90 5.62 5.30 6.86 5.72 5.88
300 500 700 900 300 300 300
5.68 5.52 5.33 5.11 6.17 6.78 6.78
8.48 8.20 7.90 7.58 8.55 9.88 9.90
3.69 3.58 3.46 3.36 4.24 4.49 4.52
8.50 8.21 7.90 7.55 8.88 9.80 9.77
Exp tbsma eam Ni
Q-SC
Exp tbsma eam
The first four rows for Pd and Ni are the calculations using QSC at various temperatures. Our results at 300 K are compared with the experimental (Exp) [54] and the other two potential results: tight binding second moment approximation (tbsma) [17] and embedded atom model (eam) [55,56].
The results obtained from the homogeneous deformation method for pure metals at 0 K and alloys at 300 K are given in Table 5, along with comparison with experimental data, whenever available, and the results of previous works using different potential models. As shown, the potential used by Cleri and Rosato [17] for the elastic constants is yielding better results. The elastic constants and bulk modulus both from Q-SC and SC calculations agree with the available experiment and the other calculations except for elastic constant of C 44 : The only metal alloy studied here for which the first-principle calculations are available is Pd. While the accuracy of the Q-SC elastic constant of C 11 for Pd, showing a deviation of 8% from experiment, is comparable to that of the embedded atom model (eam) [59] and firstprinciple calculations (LAPW) [60], SC elastic constant of C 12 is in good agreement with experiment. The experimental data on the elastic constants for Pd–Ni alloys are not available for comparison. Our results for Pd0.45Ni0.55 are smaller than the values calculated by Upadhyaya et al. [25] using the TMMP of Animalu [28] except
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Table 4 Elastic constants of Pd and Ni at 300 K in the units of GPa Metal
Els. Cons.
Exp
Q-SC
SC
H
L
DHL ð%Þ
H
L
DHL ð%Þ
Pd
C 11 C 12 C 44
227.1 176.0 71.7
201.89 140.15 81.44
186.71 125.67 79.78
7.1 10.3 2.0
229.48 164.45 82.72
214.21 149.74 81.29
6.7 8.9 1.7
Ni
C 11 C 12 C 44
250.8 150.0 123.5
214.57 152.95 89.04
237.42 164.53 105.93
10.7 7.6 19.0
213.69 166.10 69.91
237.55 180.34 83.36
11.2 8.6 19.2
The calculated results (with Q-SC and SC) are obtained by homogeneous deformations ðHÞ and long waves ðLÞ methods. Experimental values are taken from Ref. [58]. Table 5 Comparison of calculated and experimental (Exp) [58] values for elastic constants (C ij ) and bulk modulus ðBÞ at 0 K for pure metals and at 300 K for Pd–Ni alloys in the units of GPa Metal
Model
C 11
C 12
C 44
B
Pd
Exp Q-SC SC eam tbsma LAPW
234.12 216.00 248.20 218.00 232.00 218.00
176.14 150.25 175.90 184.00 178.00 172.00
71.17 91.66 93.29 65.00 73.00 74.00
195.00 172.17 200.00 195.00 196.00
Pd0.8Ni0.2 Pd0.6Ni0.4
Q-SC Q-SC
198.83 197.63
145.69 150.64
79.37 77.83
163.02 165.75
Pd0.45Ni0.55
Q-SC SC TMMP
199.13 214.97 224.50
153.29 174.33 154.70
77.69 69.34 94.50
168.57 187.88
Pd0.4Ni0.6 Pd0.2Ni0.8
Q-SC Q-SC
199.28 204.81
152.54 154.37
78.20 82.26
168.00 170.78
Ni
Exp Q-SC SC eam tbsma
261.20 219.59 226.61 233.00 257.00
150.80 165.34 178.77 154.00 155.00
131.70 99.57 79.24 128.00 136.00
188.00 183.62 194.73 180.40 189.00
Our results (Q-SC and SC) for Pd and Ni are compared with the other potential models: embedded atom method (eam) [59], tight binding second moment approximation (tbsma) [17], firstprinciples full potential (LAPW) [60], where available. A comparison for Pd0.45Ni0.55 is made between our results of QSC and SC calculations and transition metal model potential (TMMP) [25].
for C 12 calculated from SC calculations. We are also interested in temperature dependence of elastic constants in order to see whether there is
an improvement of Q-SC on the results. As the temperature increases, the Q-SC parameters improve the elastic constants slightly, as shown in Table 6. Thermal softening of alloy occurs with increasing temperature.
4. Conclusion We have presented MD simulations of the lattice dynamics of disordered Pd–Ni metal alloys by using Q-SC and SC potential parameters in this study. The transferability of the potential is an important conclusion which can be made from this work. Although the potential parameters were fitted to solid experimental properties of the pure system, the Q-SC model describes the lattice properties of Pd–Ni alloys. The contributions up to the 8th nearest-neighbor shell to force constants are found sufficient to achieve convergence for the phonon dispersion calculations. Taking into account the other Nbody potentials (see Table 3), Q-SC potential seems to give quite an accurate description of phonon properties. Even though Q-SC potential is fitted to phonon frequencies at only X point, the overall structure of dispersion curves are well reproduced. This potential produces better results for the transverse phonon frequencies than those obtained by pair potentials [25–27]. We may also concluded that SC potential agrees with experimental results despite fitting not to phonon frequencies. This work also presents the phonon
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Table 6 Comparison of calculated (Q-SC and SC) and experimental (Exp) [58] values for elastic constants (C ij ) of Ni in the units of GPa at various temperatures T (K)
0 300 500 700
Exp
Q-SC
SC
C 11
C 12
C 44
C 11
C 12
C 44
C 11
C 12
C 44
261.20 250.80 239.60 226.10
150.80 150.00 149.00 147.30
131.70 123.50 115.90 107.90
216.33 214.57 203.87 193.34
165.337 152.95 147.53 141.75
99.39 89.04 82.79 76.35
251.736 213.69 201.73 188.20
191.12 166.10 158.84 151.21
89.44 69.91 63.58 56.65
dispersion curves of alloys by using the SC potentials for the first time. Elastic constants for Pd–Ni alloys, except for Pd and Ni, are also calculated by using many-body potential for the first time by this study. Two methods have been used; homogeneous deformation and long waves. Our results from both methods are comparable for each potential parameters. Hence the consistency condition of matching the elastic constants calculated by two methods are obtained. We have seen that Q-SC potential improves the elastic constants slightly as the temperature increases. The experimental data on the elastic constants for Pd–Ni alloys except for Pd and Ni are not available for comparison. These data may encourage the experimentalists to verify our results. In summary, our work shows that Q-SC potential may be used to predict the lattice dynamics of FCC transition metals and their alloys with sufficient accuracy and to provide a detailed understanding of phonon properties. This study may give a helpful guide to researchers to improve the potential parameters for real systems. The simulation results may be further improved by fitting to experimental or first-principle calculation of solid properties of the alloy.
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