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

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 (4–6) 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 (8–10). The elastic constants and dielectric constants (11), physical and natural properties of platinum have been elucidated by expedient of theoretical models (12–16), 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 (21–23) 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.

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.

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.

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 E_F, 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.

The specific heat and Debye temperature Θ_D have been calculated as a function of temperature T from the lattice frequency spectra as shown in Figure 3. Debye temperature Θ_D 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.

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 Θ_D have been compared with available experimental data (27–29) and the curve for Θ_D 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 Θ_D 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 (Θ_D–T) curve for platinum has given excellent agreement with the experimental values (8–10). 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.

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 (8–10) 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 (16–20) 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 (32–42) 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.