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Johnson Matthey Technol. Rev., 2021, 65, (1), 87

doi:10.1595/205651320x15843541021642

Lattice Dynamical Study of Platinum by use of Van der Waals Three Body Force Shell Model

Good agreement between experimental and theoretical results

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Article Synopsis

The present article considers the lattice dynamical study of platinum by use of the Van der Waals three body force shell model (VTBFSM) due to high stiffness constant C11 and C12. The model uses the frequencies of the optical and vibrational branches in the direction [100] and phonon density of states (DOS). The study of phonon spectra is important in determining the mechanical, electrical and thermodynamic properties of elements and their alloys. The present model incorporates the effect of Van der Waals interactions (VWI) and three-body interactions (TBI) into the rigid shell model (RSM) with face-centred cubic (fcc) structure, operative up to the second neighbours in short range interactions. The available measured data for platinum agrees well with our results.

Introduction

The lattice dynamical study of metallic crystals is an interesting field of research. Platinum group metals are highly valuable transition metals which have many useful properties. The electronic structure of the platinum metals is of impressive theoretical and practical importance. Dependable thermodynamic information is expected to give the crude material from which lattice dynamics, electronic conveyances and energy states can be deduced by genuine understudies of the solid state. In the present manuscript the author has used VTBFSM for theoretical calculation. The pioneering work of Kellerman (1) for ionic interactions in the alkali halides has attracted considerable attention theoretically as well as experimentally. Löwdin’s (2, 3) and Lundqvist’s (46) theory for ionic solids leads to the first important term as many body force which includes the three-body component. The Heitler-London theory and the free-electron approximations will employ the combined effects of VWI and TBI in RSM (7). The effects of VWI and TBI in the framework of ion polarisable RSM (IPRSM) are effective up to the second neighbour with short-range, VWI and TBI interaction. The experimental investigation for the phonon dispersion curve of platinum has been done with coherent inelastic neutron scattering, variation in Debye temperature and Raman spectra (810). The elastic constants and dielectric constants (11), physical and natural properties of platinum have been elucidated by expedient of theoretical models (1216), which has also successfully described their interesting properties. After the failure of the Kellerman rigid-ion model (RIM) then Karo and Hardy (17) used a deformation dipole model, Woods et al. (7) and Dick and Overhauser (18) used a RSM to report lattice properties of alkali halides. The other most prominent model was also proposed by some researchers, among them Schröder’s (19) breathing shell model, Basu and Sengupta’s (20) deformable shell model and the three-body force shell model of Verma and Singh (2123) and Singh et al. (24) for such halides. In consideration of the effect of VWI, reported by Upadhyaya et al. (25), excellent results have been procured between experiment and theory for ionic halides and semiconductors. The betterment of the present model VTBFSM over others can be realised from the fact that in the present model relatively fewer numbers of parameters have been able to interpret numerous and largely divergent physical properties of materials. This has to motivate the author to incorporate this model in the present study.

Theory

The VWI potential originates with the correlation of the motions of an electron in various atoms, due to which shifting in electrons of the atom has occurred with respect to the nucleus and accordingly an atom becomes an electric dipole. The present model thus consists of the long-range screened Coulomb, VWI, TBI and the short-range lap aversion viable up to the second-neighbour ions in platinum. In metals, because of angular interaction between electron gas and pair of ions, the Cauchy discrepancy arises. The significant equation for the crystal potential per unit cell can be derived with VTBFSM and is given as Equation (i):

(i)

where ΦC is long-range Coulomb interaction potential. For infinite lattice in a crystal the Coulomb potential energy is given as Equation (ii):

(ii)

where αm and r0 are the Madelung constant and equilibrium nearest neighbours distance. The methodical expressions in terms of negative exponential power laws for the repulsive energy are given as Equation (iii):

(iii)

where, a (or b) and η (or ρ) are called the strength and hardness Born parameters and ΦR is a short-range lap repulsion potential. The third term ΦTBI long-range TBI interaction potential is expressed as Equation (iv):

(iv)

where f(r)0 is the equilibrium electron wave-functions. Since we consider only one ion to be polarisable and deformable, the basic equations of Singh and Verma’s (2123) model are modified. The secular determinant equation is given by Equation (v):

(v)

Here D (q) is the (6 × 6) driving matrix for RSM. The dipole-dipole (VWI) energy up to the second neighbour is expressed as Equation (vi):

(vi)

where Sv is the lattice sum and C++ and C–– are the constants ion pairs respectively. By use of the secular Equation (v) the expressions for elastic constants can be derived and given as Equations (vii)(ix):

(vii)
(viii)
(ix)

at equality condition [(d Φ/dr )0=0] we get Equations (x) and (xi):

(x)
(xi)

The frequency distribution function by use of Debye’s model is given by Equation (xii):

(xii)

To determine the phonon DOS for each polarisation is given by Equation (xiii):

(xiii)

where N is a normalisation, K is wave vector and L3 = V. The value of g(ω) is the ratio of the eigenstates number in the (ω, ω + dω) frequency interval to the total number of eigenstates jth normal mode frequency ωj(q) i.e. phonon wave vector q such that ∫g(ω) = 1.

Numerical Computations

In the present paper the parameters including (C11, C12 and C44), polarisabilities (α1, α2), and lattice constant by (26) have been theoretically calculated for platinum and given in Table I. By solving Equations (i) and (v) we can obtain the phonon spectra in the first Brillouin zone divided into evenly spaced miniature cells. The theoretical results were obtained by VTBFSM. We have used the computed vibration spectra to study the specific heat and infrared (IR)/Raman spectra in the present paper. The DOS have been obtained by computing the DOS of the frequencies from the knowledge of lattice vibrational frequency spectra. The values of frequencies are compatible with theoretical and experimental peaks and Cauchy-discrepancy for lattice dynamics of platinum.

Table I

Cauchy-Discrepancy and Constant Parameters of Platinum

Input data for platinum Calculated input parameters for platinum
Properties Values for platinum Properties Values for platinum
C11 34.67 (26) C11 32.57
C12 25.07 (26) C12 23.97
C44 7.65 (26) C44 6.35
2a 3.923 (26) 2a 3.923
α1 0.037 α1 0.0487
α2 0.032 α2 0.0368

In the Brillouin zone surface the calculated phonon dispersion curves for platinum are shown in Figure 1 by using first principles along two high symmetry directions (qqq) Г-X-Г. The parallel vibrational modes show real dispersion with a maximum cleave. The upper branch consists of longitudinal modes, while the lower one is the shear‐horizontal mode, along both the Г-X-Г directions. We find that the surface modes for clean platinum (qqq) undergo a few changes in L-T modes on the clean surface, near the zone boundaries and along the X-direction, are replaced only in the dispersion curves. This is because the zone boundaries are moderate and the next two surface modes are strengthened. The experimental reported results for dispersion relation are shown in Figure 1(b) (27). On comparing with the experimental result i.e. Figure 1(a) with Figure 1(b) good agreement can be observed.

Fig. 1.

(a) Phonon dispersion relation curve for platinum by VTBFSM; (b) phonon dispersion relation to platinum along the [ɛ, 0, 0] direction (A) and [ɛ, ɛ, 0] direction (B) (27). Copyright © 2008 Società Italiana di Fisica. Reprinted by permission of Springer Nature

(a) Phonon dispersion relation curve for platinum by VTBFSM; (b) phonon dispersion relation to platinum along the [ɛ, 0, 0] direction (A) and [ɛ, ɛ, 0] direction (B) (27). Copyright © 2008 Società Italiana di Fisica. Reprinted by permission of Springer Nature

The DOS vs. frequency curve for platinum theoretically calculated in the energy range from approximately –7.0 eV to 0.5 eV in bulk is shown as a solid line in Figure 2. The bulk DOS exhibits three main peaks that are accurately produced. The small observed differences are related to the shape of the main peaks. The differences between the calculated and observed values were found and discussed in Figure 2. There are very important features obtained from the DOS vs. frequency curve. The information about the surface and resonance states was found through these differences. Below the Fermi level EF, resonance-states are expected to be obtained, mainly because these energies represent the continuum and few energy gaps exist at these energy values. The experimental reported DOS curve (28) may be compared with the present model i.e. Figure 2(a). Sharp peaks can be seen in Figure 2(a) while in Figure 2(b) the distortion in peaks can be seen which justifies the superiority of the present theoretical study.

Fig. 2.

(a) DOS vs. frequency curve of platinum with present VTBFSM; (b) DOS vs. frequency curve of platinum. Reprinted from (28), with the permission of AIP publishing

(a) DOS vs. frequency curve of platinum with present VTBFSM; (b) DOS vs. frequency curve of platinum. Reprinted from (28), with the permission of AIP publishing

The specific heat and Debye temperature ΘD have been calculated as a function of temperature T from the lattice frequency spectra as shown in Figure 3. Debye temperature ΘD is calculated from different frequency values. The specific heat value of platinum has been measured at extended temperature (0 K to 300 K). The calculated result is in reasonable agreement at moderate temperature and at very low temperature. The comparison can be seen through experimental results (27).

Fig. 3.

(a) Debye temperature curve with VTBFSM; (b) Debye temperature curve by Closs and Shukla (27) Copyright © 2008 Società Italiana di Fisica. Reprinted by permission of Springer Nature

(a) Debye temperature curve with VTBFSM; (b) Debye temperature curve by Closs and Shukla (27) Copyright © 2008 Società Italiana di Fisica. Reprinted by permission of Springer Nature

Discussion

The varying investigated properties are distinctly shown in the present study by successful use of VTBFSM, which has provided the complete lattice description of platinum. It agrees well with the test of anisotropy factor A = 2C44/(C11–C12) > unity and towards the high frequency end the higher peak is found. The determined model parameters in Table I were used to solve the secular equation for specified values of wave vectors in the first Brillouin zone, which is split up in an evenly spaced sample of (1000) wave vectors by Kellerman (1). From the symmetry, these 1000 points are reduced to 48 non-equivalent points at which the vibration frequencies have been obtained by solving the secular determinant. Debye temperature variations at different temperatures by Macfarlane et al. (26) and colossal dielectric constant (CDC) curves for platinum crystals have been computed by using VTBFSM model. By using the sampling technique the corresponding values of ΘD have been compared with available experimental data (2729) and the curve for ΘD vs. absolute temperature (T) was plotted as shown in Figure 1 for platinum. In this temperature range one should take into account the temperature dependence of the frequency spectrum, this requires, however, knowledge of the phonon frequencies at more than one temperature. The variations of ΘD with specific heats have been used to compute phonon frequencies in the first Brillouin zone and data points for different points were reported in Table II. The effect of anharmonicity is excluded so slight discrepancies between theoretical and experimental results at higher temperatures are seen, though the agreement is almost better with VTBFSM. The calculated (ΘD–T) curve for platinum has given excellent agreement with the experimental values (810). The DOS curve is given in Figure 2 and data points are shown in Table III. The two-phonon Raman spectra are sensitised to the high-frequency side while the specific heats are sensitive to its lower side which is stated the reasonableness of VTBFSM for all wavelength range.

Table II

Assignments for the Observed Peak Positions for Phonon Dispersion Relation in Г-X-Г Direction

Г-directions X-directions Г-directions
x-axis y-axis x-axis y-axis x-axis y-axis
0.05283 1.21564 0.4084 5.1927 1.50882 6.4594 2.51486 0.2413
0.09694 1.29258 0.8167 4.3248 1.54589 4.4531 2.58016 0.0742
0.14104 1.29749 1.2251 4.2535 1.54593 4.4548 2.5892 0.3527
0.16361 1.29889 1.9118 4.2383 1.61352 6.3852 2.61588 0.7981
0.1895 1.38452 1.6334 4.5661 1.64652 4.6218 2.633 0.4826
0.19903 1.39083 2.3573 4.4389 1.65054 4.3063 2.67354 1.6334
0.22918 1.39743 1.9861 4.3991 1.67279 4.6961 2.67698 0.7796
0.23438 1.49138 2.7471 6.478 1.70493 6.1439 2.70459 2.0789
0.27413 3.174 1.74195 4.065 2.73415 1.1694
0.27765 2.3944 1.75174 5.0302 2.78695 1.5406
0.30512 3.5638 1.78317 5.8283 2.84847 1.9118
0.33485 2.8028 1.82898 4.9924 2.91871 2.2645
0.34487 3.9907 1.8306 5.29 2.98776 5.5313
0.389 4.4176 1.85263 5.4756 2.99327 2.58
0.39202 3.1926 1.87869 5.4529 3.04495 5.9397
0.42283 3.4153 1.87879 5.4385 3.08086 2.8399
0.42875 4.8445 1.90724 3.5081 3.09402 2.8956
0.47724 5.2715 1.94376 4.9745 3.11517 6.2738
0.48872 3.7865 1.97845 3.8125 3.19451 2.9698
0.53007 5.6798 1.97928 3.6312
0.58951 4.1392 1.98546 3.174
0.59596 6.051 2.03477 4.3619
0.67489 6.3666 2.05494 2.8399
0.68143 4.3619 2.07071 5.29
0.75796 6.478 2.0911 2.8566
0.81252 4.4733 2.09835 2.6172
0.85822 6.348 2.14018 4.9374
0.86919 4.3991 2.14743 3.5824
0.90881 4.6961 2.14883 4.8631
0.95402 6.1253 2.16345 2.2645
0.97385 4.2877 2.19499 3.1555
0.9964 4.9559 2.20077 4.4548
1.03665 5.8283 2.22853 1.8933
1.07096 5.2715 2.24394 4.0093
1.07858 4.2506 2.25891 1.7262
1.1149 5.5313 2.28713 3.5824
1.13663 5.5154 2.32159 3.1555
1.13671 5.5128 2.32399 1.355
1.18774 4.2691 2.35607 2.7471
1.21487 5.123 2.38907 2.2744
2.39051 2.3016
2.44978 0.6125
Table III

Assignments for the Observed Peak Positions for Combined Density of States

x-axis, THz y-axis, arbitrary units x-axis, THz y-axis, arbitrary units
0.0234 3.1403 0.01226 0.23887 1.2142 0.50083
0.1286 3.2455 0.05927 0.28588 1.2586 0.44984
0.2224 3.3964 0.10831 0.32272 1.2914 0.39884
0.3276 3.51 0.15532 0.27586 1.3686 0.29481
0.4213 3.6352 0.20437 0.229 1.4015 0.24381
0.562 3.7489 0.27997 0.18418 1.4459 0.16746
0.6212 3.8852 0.32899 0.12713 1.4788 0.14386
0.692 4.1503 0.38006 0.13548 1.8082 0.00327
0.7512 4.4271 0.42908 0.14791 2.0736 0.01978
0.8219 4.6924 0.48015 0.16033 2.3276 0.03832
0.8811 4.9576 0.52917 0.17072 2.5589 0.06909
0.9518 4.9918 0.5782 0.16054 2.7902 0.09782
1.0111 5.1402 0.62926 0.11778 2.8599 0.11827
1.104 5.2884 0.65177 0.07299 2.9651 0.16528
1.1369 5.4367 0.60077 0.03023 3.0818 0.2123
1.1812 0.54978

Conclusion

The exploration of model parameters, Debye temperature and DOS are reported by use of the present model VTBFSM for platinum. The conformity with experimental data (810) of our theoretical peak is very good for platinum. A successful explanation of spectra has provided the next best test of any model for their higher range of frequency. Small deviations were observed at the higher temperature side due to harmonic approximation in the Debye curve. Better agreement has been obtained with the available experimental data (1620) and theoretical results. The motivation of this work is the availability of experimental (29) and theoretical (30, 31) work on platinum. Therefore, it may be concluded that the incorporation of VWI is requisite for the absolute interpretation of the phonon dynamical behaviour of platinum. Many researchers have also successfully reported theoretical results for alkali halides (3242) by use of the present model. Hence, the present model may be understood to provide a powerful and simple approach for a comprehensive study of the harmonic as well as anharmonic elastic properties of the crystals under consideration. The only constraint of VTBFSM is the knowledge of certain experimental parameters needed that can be used as input data.

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The Author


U. C. Srivastava obtained his MSc (Physics) and PhD (Solid State Physics) degrees from Veer Bahadur Singh Purvanchal University, Jaunpur, Uttar Pradesh, India. He obtained his MTech (Electronics & Telecommunication) degree from Karnataka State Open University, Mysore, India. He currently works as Assistant Professor-III in the Department of Physics, Amity Institute of Applied Sciences, Amity University, India. He has 17 years’ teaching and research experience. His area of research is theoretical lattice dynamical study of ionic crystals. He has published 24 research papers in different national and international journals.

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