Vibrational dynamics of Ni-glassy alloys

Abstract

The well known model potential is used to investigate the vibrational properties of some Ni-based binary glassy alloys using three theoretical models. Different local field correction functions are employed to see the effect of exchange and correlation in the aforesaid properties and have been found successful.

Keywords

pseudopotential / pair potential / phonon dispersion curves (PDC) / Ni-based amorphous binary alloys / local field correction functions

Introduction

The mankind has been manufacturing glassy materials for several thousand years. Compared to that, the scientific study of amorphous materials has a much shorter history. And only recently, there has been an explosion of interest to these studies as more promising materials are produced in the amorphous form. The range of applications of metallic glasses is vast and extends from the common window glass to high capacity storage media for digital devices [1-23].

The Ni₃₁Dy₆₉ glass is a member of the transition metal-lanthanide or actinide (TM-LA) element group, while Ni₈₁B₁₉ is a candidate of the transition metal-metalloid (TM-M) glass. Another four metallic glasses such as Ni₃₃Y₆₇, Ni₃₆Zr₆₄, Ni₅₀Zr₅₀ and Ni₆₀Nb₄₀ are members of the transition metallic (TM-TM) glasses. The phonon dynamics of Ni₃₃Y₆₇ glass was studied by Arun Pratap et al. [1] using Hubbard-Beeby (HB) [3] and Takeno-Goda (TG) [4,5] approaches. The vibrational dynamics of Ni₅₀Zr₅₀ glass has been studied by Gupta et al. [2] using Takeno-Goda (TG) [4,5] and Bhatia-Singh (BS) [5] approaches. The phonon dynamics of Ni₃₁Dy₆₉, Ni₃₆Zr₆₄, Ni₆₀Nb₄₀ and Ni₈₁B₁₉ metallic glasses is not reported previously using model potential formalism. Recently, we have reported the vibrational properties of some binary metallic glasses [8-23].

Looking to the advantages of metallic glasses, the present paper is emphasizing the phonon dynamics of six Ni-based amorphous alloys viz. Ni₃₁Dy_69, Ni₃₃Y₆₇, Ni₃₆Zr₆₄, Ni₅₀Zr₅₀, Ni₆₀Nb₄₀ and Ni₈₁B₁₉ using well recognized model potential [8-23]. The thermodynamics and elastic properties such as longitudinal sound velocity {\upsilon }_L, transverse sound velocity {\upsilon }_T, isothermal bulk modulus B_T, modulus of rigidity G, Poisson’s ratio \sigma, Young’s modulus Y and Debye temperature {\theta }_{\mathrm{D}} are computed from the elastic limit of the dispersion relation. Five different types of local field correction functions proposed by Hartree (H) [24], Taylor (T) [25], Ichimaru-Utsumi (IU) [26], Farid et al. (F) [27] and Sarkar et al. (S) [28] are used to study the exchange and correlation effects in the aforesaid studies. The phenomenological theory of Hubbard-Beeby [3] in the random phase approximation is employed to generate the phonon dispersion curves (PDC). The most important ingredient of the PDC is pair potential computed theoretically in Wills-Harrison (WH) [29] form from the well recognized model potential [8-23].

Theoretical methodology

The fundamental ingredient, which goes into the calculation of the phonon dynamics of metallic glasses, is the pair potential. In the present study, for TM-LA, TM-TM and TM-M metallic glasses, the pair potential is computed using [6-23,29],

(1)V(r)=V_S(r)+V_b(r)+V_r(r)

The s-electron contribution to the pair potential V_S(r) is calculated from [6-23]

(2)V_s(r)={\displaystyle \frac{Z_S^{\mathrm{2}}{\mathrm{e}}^{\mathrm{2}}}{r}}+{\displaystyle \frac{{\mathrm{\Omega }}_O}{{\mathrm{\pi }}^{\mathrm{2}}}}{\displaystyle \int F(q){\displaystyle \frac{\sin (qr)}{qr}}{q}^{\mathrm{2}}\mathrm{d}q}

Here Z_S~1.5 is found by integrating the partial s-density of states resulting from self-consistent band structure calculation for the entire 3d and 4d series [29], while {\mathit{\Omega }}_O is the effective atomic volume of the one component fluid.

The energy wave number characteristics appearing in Eq. (2) is written as [6-23,29]

(3)F(q)={\displaystyle \frac{-{\mathit{\Omega }}_O{q}^{\mathrm{2}}}{\mathrm{16}\mathrm{\pi }}}{\left|W_B(q)\right|}^{\mathrm{2}}{\displaystyle \frac{{ϵ}_H(q)-\mathrm{1}}{\mathrm{1}+[{ϵ}_H(q)-\mathrm{1}][\mathrm{1}-f(q)]}}

where, W_B(q) is the effective bare ion potential, {ϵ}_H(q) the Hartree dielectric response function [24] and f(q) is the local field correction function to introduce the exchange and correlation effects.

The well recognized model potential W_B(q) [8-23] used in the present computation of phonon dynamics of binary metallic glasses is of the form

(4)W_B(q)={\displaystyle \frac{-\mathrm{4}\mathrm{\pi }{e}^{\mathrm{2}}Z}{{\mathit{\Omega }}_O{q}^{\mathrm{2}}}}\left[\begin{array}{l}\left\{\begin{array}{l}-\mathrm{1}+{\displaystyle \frac{\mathrm{12}}{U^{\mathrm{2}}}}+{\displaystyle \frac{U^{\mathrm{2}}}{\mathrm{1}+U^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{6}{U}^{\mathrm{2}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{18}{U}^{\mathrm{2}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{3}}}}-{\displaystyle \frac{\mathrm{6}{U}^{\mathrm{4}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{3}}}}\hfill \\ +{\displaystyle \frac{\mathrm{24}{U}^{\mathrm{2}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}-{\displaystyle \frac{\mathrm{24}{U}^{\mathrm{4}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}\hfill \end{array}\right\}\cos U\hfill \\ +\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{6}}{U}}-{\displaystyle \frac{\mathrm{12}}{U^{\mathrm{3}}}}+{\displaystyle \frac{U}{\mathrm{1}+U^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{3}U}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{3}{U}^{\mathrm{3}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{6}U}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{3}}}}\hfill \\ -{\displaystyle \frac{\mathrm{18}{U}^{\mathrm{3}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{6}U}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}-{\displaystyle \frac{\mathrm{36}{U}^{\mathrm{3}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}+{\displaystyle \frac{\mathrm{6}{U}^{\mathrm{5}}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}\hfill \end{array}\right\}\sin U\hfill \\ +\mathrm{24}{U}^{\mathrm{2}}\exp (\mathrm{1}){\displaystyle \frac{U^{\mathrm{2}}-\mathrm{1}}{{(\mathrm{1}+U^{\mathrm{2}})}^{\mathrm{4}}}}\hfill \end{array}\right].

here U=q{r}_C·r_C is the model potential parameter. This form has the feature of a Coulombic term outside the core and varying cancellation due to repulsive and attractive contributions to the potential within the core in real space. The detailed information of this potential is given in the literature [8-23]. The model potential parameter {\mathit{r}}_{\mathit{C}} is calculated from the well known formula [8-23] as follows:

(5)r_C={\displaystyle \frac{\mathrm{0.51}{r}_S}{Z^{\mathrm{1}/\mathrm{3}}}}

Here r_S is the Wigner-Seitz radius of the amorphous alloys.

The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z_d, the d-state radii r_d and the nearest-neighbor coordination number N_C as follows [29]:

(6)V_b(r)=-Z_d\left(\mathrm{1}-{\displaystyle \frac{Z_d}{\mathrm{10}}}\right){\left({\displaystyle \frac{\mathrm{12}}{N_C}}\right)}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.}{\displaystyle \frac{\mathrm{28.06}}{\mathrm{\pi }}}·{\displaystyle \frac{\mathrm{2}{r}_d^{\mathrm{3}}}{r^{\mathrm{5}}}},

and

(7)V_r(r)=Z_d{\displaystyle \frac{\mathrm{450}}{{\mathrm{\pi }}^{\mathrm{2}}}}·{\displaystyle \frac{r_d^{\mathrm{6}}}{r^{\mathrm{8}}}}.

The theories of Hubbard-Beeby [3], Takeno-Goda [4,5] and Bhatia-Singh [6,7] have been employed in the present computation. The expressions for longitudinal phonon frequency {\omega }_L and transverse phonon frequency {\omega }_T as per HB, TG and BS approaches are given below [3-7].

According to the HB [3], the expressions for longitudinal phonon frequency {\omega }_L and transverse phonon frequency {\omega }_T are [3]

(8){\omega }_L^{\mathrm{2}}(q)={\omega }_E^{\mathrm{2}}\left[\mathrm{1}-{\displaystyle \frac{\sin (q\sigma )}{q\sigma }}-{\displaystyle \frac{\mathrm{6}\cos (q\sigma )}{{(q\sigma )}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{6}\sin (q\sigma )}{{(q\sigma )}^{\mathrm{3}}}}\right]

and

(9){\omega }_T^{\mathrm{2}}(q)={\omega }_E^{\mathrm{2}}\left[\mathrm{1}-{\displaystyle \frac{\mathrm{3}\cos (q\sigma )}{{(q\sigma )}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{3}\sin (q\sigma )}{{(q\sigma )}^{\mathrm{3}}}}\right]

with {\omega }_E^2={\displaystyle \frac{4\mathrm{\pi }\rho }{3M}}{\displaystyle \underset{0}{\overset{\infty }{\int }}g\left(r\right)V^{\prime \prime }\left(r\right)r^2\mathrm{d}r} is the maximum frequency.

The theory for computing the phonon dynamics in amorphous solids, proposed by TG [4,5], has been employed in the present computation. The expressions for longitudinal phonon frequency {\omega }_L and transverse phonon frequency {\omega }_T as per TB approach are [4,5]

(10){\omega }_L^{\mathrm{2}}(q)={\displaystyle \frac{\mathrm{4}\mathrm{\pi }\rho }{M}}{\displaystyle \underset{\mathrm{0}}{\overset{\infty }{\int }}\mathrm{d}rg(r)\left\{\begin{array}{l}r{V}^{\prime }(r)\left(\mathrm{1}-{\displaystyle \frac{\sin (qr)}{qr}}\right)+[r^{\mathrm{2}}{V}^{″}(r)-r{V}^{\prime }(r)]\hfill \\ \left({\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}-{\displaystyle \frac{\sin (qr)}{qr}}-{\displaystyle \frac{\mathrm{2}\cos (qr)}{{(qr)}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{2}\sin (qr)}{{(qr)}^{\mathrm{3}}}}\right)\hfill \end{array}\right\}}

and

(11){\omega }_T^{\mathrm{2}}(q)={\displaystyle \frac{\mathrm{4}\mathrm{\pi }\rho }{M}}{\displaystyle \underset{\mathrm{0}}{\overset{\infty }{\int }}\mathrm{d}rg(r)\left\{\begin{array}{l}r{V}^{\prime }(r)\left(\mathrm{1}-{\displaystyle \frac{\sin (qr)}{qr}}\right)+[r^{\mathrm{2}}{V}^{″}(r)-r{V}^{\prime }(r)]\hfill \\ \left({\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}+{\displaystyle \frac{\mathrm{2}\cos (qr)}{{(qr)}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{2}\sin (qr)}{{(qr)}^{\mathrm{3}}}}\right)\hfill \end{array}\right\}}

Here M and \rho are the atomic mass and the number density of the glassy alloy, respectively, while V^{″}(r) is the second derivative of the pair potential.

Recently BS [6] was modified by Shukla and Campnaha [7]. They introduced screening effects in the BS approach. Then, with the above assumptions and modification, the dispersion equations for an amorphous material can be written as [6,7]

(12)\rho {\omega }_L^{\mathrm{2}}(q)={\displaystyle \frac{\mathrm{2}{N}_C}{q^{\mathrm{2}}}}(\beta I_{\mathrm{0}}+\delta I_{\mathrm{2}})+{\displaystyle \frac{k_e{k}_{TF}^{\mathrm{2}}{q}^{\mathrm{2}}ϵ(q){\left|G(q{r}_S)\right|}^{\mathrm{2}}}{q^{\mathrm{2}}+k_{TF}^{\mathrm{2}}ϵ(q)}}

and

(13)\rho {\omega }_T^{\mathrm{2}}(q)={\displaystyle \frac{\mathrm{2}{N}_C}{q^{\mathrm{2}}}}\left(\beta I_{\mathrm{0}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\delta (I_{\mathrm{0}}-I_{\mathrm{2}})\right)

The other details of used constants in the BS approach were already narrated in the literatures [23,24]. Here M is the effective atomic mass, \rho is the effective number density, and N_C the effective coordination number of the glassy system.

In the long wavelength limit of the frequency spectrum, both phonon frequencies viz. the longitudinal {\omega }_L and transverse {\omega }_T phonon frequencies are proportional to the wave vectors and obey the relationships [8-23]

{\omega }_L\propto q\mathrm{a}\mathrm{n}\mathrm{d}{\omega }_T\propto q,

(14){\omega }_L={\upsilon }_{L}q\mathrm{a}\mathrm{n}\mathrm{d}{\omega }_T={\upsilon }_{T}q

where, {\upsilon }_L and {\upsilon }_T are the longitudinal and transverse sound velocities of the glassy alloys, respectively. Detailed expressions of the long wavelength limit of the frequency spectrum are narrated in our earlier papers.

The present study also includes isothermal bulk modulus B_T, modulus of rigidity G, Poisson’s ratio \sigma, Young’s modulus Y and Debye temperature {\theta }_D from the elastic limit of the PDC. All the quantities are computed from the longitudinal and transverse sound velocities ({\upsilon }_L and {\upsilon }_T). The bulk modulus B_T, modulus of rigidity G, Poisson’s ratio \sigma, Young’s modulus Y and Debye temperature {\theta }_D are obtained using the expressions [3-23],

(15)B_T={\rho }_M\left({\upsilon }_L^{\mathrm{2}}-{\displaystyle \frac{\mathrm{4}}{\mathrm{3}}}{\upsilon }_T^{\mathrm{2}}\right)

(16)G={\rho }_M{\upsilon }_T^{\mathrm{2}}

with {\rho }_M being the isotropic number density of the solid.

(17)\sigma ={\displaystyle \frac{\mathrm{1}-\mathrm{2}{\displaystyle \frac{{\upsilon }_T^{\mathrm{2}}}{{\upsilon }_L^{\mathrm{2}}}}}{\mathrm{2}-\mathrm{2}{\displaystyle \frac{{\upsilon }_T^{\mathrm{2}}}{{\upsilon }_L^{\mathrm{2}}}}}}

(18)Y=\mathrm{2}G(\sigma +\mathrm{1})

and

(19){\theta }_{\mathrm{D}}={\displaystyle \frac{\hslash {\omega }_{\mathrm{D}}}{k_{\mathrm{B}}}}={\displaystyle \frac{\hslash }{k_{\mathrm{B}}}}\mathrm{2}\mathrm{\pi }{\left({\displaystyle \frac{\mathrm{9}\rho }{\mathrm{4}\mathrm{\pi }}}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}}{\left({\displaystyle \frac{\mathrm{1}}{{\upsilon }_L^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{2}}{{\upsilon }_T^{\mathrm{3}}}}\right)}^{-\frac{\mathrm{1}}{\mathrm{3}}}

The low temperature specific heat C_V can be calculated from the following expressions [30],

(20)C_V={\displaystyle \frac{{\mathit{\Omega }}_O{\hslash }^{\mathrm{2}}}{k_{\mathrm{B}}{T}^{\mathrm{2}}}}{\displaystyle \sum _{\lambda =L,T}{\displaystyle \int {\displaystyle \frac{{\mathrm{d}}^{\mathrm{3}}q}{{(\mathrm{2}\mathrm{\pi })}^{\mathrm{3}}}}{\displaystyle \frac{{\omega }_{\lambda }^{\mathrm{2}}(q)}{\left[\exp \left({\displaystyle \frac{\hslash {\omega }_{\lambda }(q)}{k_{\mathrm{B}}T}}\right)-\mathrm{1}\right]\left[\mathrm{1}-\exp \left(-{\displaystyle \frac{\hslash {\omega }_{\lambda }(q)}{k_{\mathrm{B}}T}}\right)\right]}}}}.

Here, \hslash, k_{\mathrm{B}} and {\omega }_{\mathrm{D}} are the Plank’s constant, Boltzmann’s constant, and Debye frequency, respectively.

Results and discussion

The input parameters and other related constants used in the present computations are tabulated in Table 1, which are computed from the pure metallic data and taken from the literature [29].

The presently calculated pair potentials of Ni₃₁Dy₆₉ glass are shown in Fig. 1(a), which elucidates that the inclusion of screening functions hardly changes the nature of the pair potentials, except around the first minimum. The well depth slightly increases due to the influence of various screening functions compared to H-screening. The presently obtained pair potentials of Ni₃₃Y₆₇ glass are shown in Fig. 1(b) along with the other such theoretical results [1,23]. The first zero position of pair potentials at r=r_{\mathrm{0}} under all screening functions occurs at r₀ ≈ 3.95 a.u.. Thus, the inclusion of exchange and correlations on the V(r=r_{\mathrm{0}}) is not substantial. But the well width increases compared to H-screening. It is observed that the well depth of presently computed pair potentials shifted towards the left and were as high as compared to the results of Arun Pratap et al. [1] and Hausleitner-Hafner [31]. The presently generated pair potentials of Ni₃₆Zr₆₄ glass are displayed in Fig. 1(c). The first zero for V(r=r_{\mathrm{0}}) under all screening functions occurs at r₀ ≈ 3.74 a.u.. The maximum depth in the pair potentials is obtained for S-function. The pair potentials under remaining T-, IU- and F-screening functions are lying between those under H- and S-screening function. The presently calculated pair potentials of Ni₅₀Zr₅₀ glass are observed in Fig. 1(d) along with the other such available theoretical data [2,32]. The position of pair potentials at r=r_{\mathrm{0}} under H-function occurs at r₀ = 3.62 a.u., while the influence of screenings suppresses this zero slightly and makes it occur at r₀≤3.55 a.u.. The broad well width is seen due to the inclusion of local field correction functions. It is seen that the well depth of presently computed pair potentials shifted towards the left and were as high as compare to the outcomes of Gupta et al. [2] and Hausleitner-Hafner [32]. The computed pair potentials of Ni₆₀Nb₄₀ glass are displayed in Fig. 1(e). It is apparent from the behavior that the inclusion of screening effects increases the well width slightly compared to H–screening. The first zero for V(r=r_{\mathrm{0}}) under H occurs at r₀ = 3.38 a.u., while the inclusion of exchange and correlations suppresses this zero to r₀≤3.28 a.u.. The pair potentials of the two components of Ni₈₁B₁₉ glass are shown in Fig. 1(f). It is noticed that the first zero for V(r=r_{\mathrm{0}}) under all local field correction functions occurs around r₀ ≈ 3.26 a.u.. The well width also increases with respect to H-screening. The position of V_{\min }(r) is affected by the nature of the screenings. It is observed that the well depth of presently computed potentials move towards the left as compared to that of Hausleitner and Hafner [33]. The present results of the pair potentials of all the Ni-based amorphous alloys do not show any oscillatory behavior and are almost constant in the large r-region. The presently computed pair potentials from H-, T-, IU-,, F- and S-local field correction functions for most of the amorphous alloys are overlapped with each other.

The pair potentials for the six Ni-based amorphous alloys computed using S-local field correction function is displayed in Fig. 2. In this figure, we have compared our presently computed pair potentials from S-local field correction function with each other for most of the amorphous alloys. This figure indicates the shifting of the pair potentials with respect to atomic volume {\mathit{\Omega }}_O of the amorphous alloys. The repulsive coulomb interaction and the attractive interactions represented by oscillatory nature are observed. It is also noticed that when volume {\mathit{\Omega }}_O of the glassy alloys increases ({\mathit{\Omega }}_O is more for Ni₃₃Y₆₇ glass), the potential depth rises. It means that the pair potential for Ni₃₃Y₆₇ glass shows higher depth in comparison with other metallic glasses. The potential well width of Ni₈₁B₁₉ glass shifts at lower r-values, while that of Ni₃₁Dy₆₉ glass shifts at higher r-values. The lanthanide element Dy and metalloid component B plays an important role in the nature of the pair potentials of respective amorphous alloys, i.e., for Ni₃₁Dy₆₉ glass, the repulsive interaction is more in composition with Ni₈₁B₁₉ metallic glass. All the pair potentials show the combined effect of the s- and d-electrons. Bretonnet and Derouiche [34] observed that the repulsive part of V(r) is drawn lower and its attractive part is deeper due to the d-electron effect. When we go from Ni₃₁Dy₆₉ → Ni₈₁B₁₉, the net number of d-electron r_d decreases, hence the V(r) is shifted towards the lower r-values. Therefore, the present results have supported the d-electron effect as noted by Bretonnet and Derouiche [34].

From Figs. 1 and 2, it is noted that the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions, which shows the waving shape oscillation of the potential after r ≈ 10 a.u.. Hence, the pair potentials converge towards a finite value instead of zero in repulsive region.

The phonon modes for longitudinal and transverse branches of Ni₃₁Dy₆₉, Ni₃₃Y₆₇, Ni₃₆Zr₆₄, Ni₅₀Zr₅₀, Ni₆₀Nb₄₀ and Ni₈₁B₁₉ metallic glasses calculated using HB approach with the five screening functions are shown in Figs. 3-8. The phonon eigen frequencies of Ni₃₁Dy₆₉ glass calculated using HB approach to study the screening influence are shown in Fig. 3. From this figure, it is seen that the present results of phonon modes due to H, T and F-function are lying between those due to IU and S-screenings. The first minimum in the longitudinal branch is found around 1.72 Å^-1 for H, 2.36 Å^-1 for T and IU, 2.52 Å^-1 for F and S-local field correction functions. The influence of T, IU and F on {\omega }_L is ranging from 7.9% to 11%. The S-function enhances the {\omega }_L of Ni₃₁Dy₆₉ glass by ≈ 67.8% in comparison to H–dielectric function. At q ≈ 1.0 Å^-1 point, the influence of various local field correction functions on {\omega }_T due to T, IU, F and S-screening is 19.98%, 22.03%, 20.27% and 35.99%, respectively. From Fig. 4, it is noted that the present results of PDC of Ni₃₃Y₆₇ glass due to T, IU and F are lying between those due to S and H-screening. The present results are found to be qualitatively in agreement with the available theoretical data [1]. The first depth in the longitudinal branch is observed around 1.75 Å^-1 for H, 2.62 Å^-1 for T and IU, 2.51 Å^-1 for F and 1.74 Å^-1 for S-function. The screening influence on the first maximum of {\omega }_L for T is 47.78%, for IU is 36.18%, for F is 46.06% and for S-screening is 43.14% with respect to H-screening, which does not include any exchange and correlation effects. At q ≈ 1.0 Å^-1 point, the influence on {\omega }_T due to T, IU, F and S-exchange and correlation functions is 2.75%, 17.78%, 1.15% and 15.55%, respectively, in comparison with H-dielectric function. From Fig. 5, it is evident that the inclusion of exchange and correlation effects suppresses the phonon frequencies of Ni₃₆Zr₆₄ glass. The first depth in the longitudinal branch is found around 2.72 Å^-1 for H, 2.03 Å^-1 for T, 2.85 Å^-1 for IU as well as for F and 2.02 Å^-1 for S-screening function. The effect of T, IU, F and S-screening at first maximum of {\omega }_L is about 24.91%, 67.28%, 49.49% and 33.56%, respectively, in comparison to static H-dielectric function. The screening effects observed at q ≈ 1.0 Å^-1 position in transverse branch for T, IU, F and S-functions is 9.72%, 50.47%, 51.52% and 20.18%, respectively. It is noticed from Fig. 6 that the inclusion of exchange and correlation effects suppresses the longitudinal as well as transverse phonon branches of Ni₅₀Zr₅₀ glass. The present results are found to be qualitatively in agreement with the available theoretical data [2]. The first depth in the longitudinal branch for H, T, IU, F and S-local field correction functions lies at q ≈ 2.72 Å^-1, 2.13 Å^-1, 2.85 Å^-1, 2.84 Å^-1 and 2.03 Å^-1, respectively. The screening influences at the first peak of {\omega }_L with respect to H-screening are 14.73% for T, 38.46% for IU, 39.50% for F and 37.08% for S-screening. Such influences on {\omega }_T at q ≈ 1.0 Å^-1 for T is 14.17%, for IU is 41.10%, for F is 42.11% and for S-screening is 24.21%. It is evident from Fig. 7 that the height of the first peak and the position of the first peak in the longitudinal and transverse modes of Ni₆₀Nb₄₀ glass are appreciably influenced by different screenings. The first minimum in the longitudinal branch for H, T, IU, F and S-local field correction functions occurs at q ≈ 2.32 Å^-1, 2.22 Å^-1, 2.21 Å^-1, 2.36 Å^-1 and 2.05 Å^-1, respectively. The effect of the T-screening at the first peak of {\omega }_L is 3.65% with respect to H-function. Such effect of IU, F and S-function is 3.65%, 2.10% and 16.44%, respectively. The screening influence on {\omega }_T at q ≈ 1.0 Å^-1 point due to T-dielectric function is 2.71%, due to IU is 17.75%, due to F-function is 1.10% and due to S is 15.58% with respect to H-dielectric function. It is seen from Fig. 8 that the inclusion of exchange and correlation effects enhances the phonon frequencies of Ni₈₁B₁₉ glass in longitudinal as well as transverse branches except for f(q) of S-function. The first minimum in the longitudinal branch for H, T, IU, F and S-local field correction functions is around q ≈ 3.35 Å^-1, 3.32 Å^-1, 3.31 Å^-1, 3.34 Å^-1 and 3.45 Å^-1, respectively. At the first maximum, the screening influences on {\omega }_L is 0.48% to 11.25% with respect to static H-screening, while such influence on {\omega }_T at q ≈ 1.0 Å^-1 point is about 14.78% to 63.10%.

The PDC calculated from the HB, TG and BS approaches with S-local field correction function of Ni₃₁Dy₆₉, Ni₃₃Y₆₇, Ni₃₆Zr₆₄, Ni₅₀Zr₅₀, Ni₆₀Nb₄₀ and Ni₈₁B₁₉ metallic glasses are shown in Figs. 9(a)–(f). The first minimum in the longitudinal branch of Ni₃₁Dy₆₉ metallic glass falls at q ≈ 1.8 Å^-1 for BS, 2.7 Å^-1 for TG and 2.6 Å^-1 for HB approach. The first minimum in the longitudinal branch of Ni₃₃Y₆₇ metallic glass falls at 1.7 Å^-1 for BS, 1.3 Å^-1 for TG and1.6 Å^-1 for HB approach. The first minimum in the longitudinal branch of Ni₃₆Zr₆₄ metallic glass falls at q ≈ 1.6 Å^-1 for BS, q ≈ 2.0 Å^-1 for TG and HB approaches. The first minimum in the longitudinal branch of Ni₅₀Zr₅₀ metallic glass occurs at q ≈ 2.0 Å^-1 for HB and TG approaches and q ≈ 2.0 Å^-1 for BS approach. The first minimum in the longitudinal branch of Ni₆₀Nb₄₀ metallic glass for HB and TG approaches falls at q ≈ 2.2 Å^-1 and for BS at q ≈ 1.6 Å^-1. The first minimum in the longitudinal branch of Ni₈₁B₁₉ metallic glass occurs for HB, TG and BS approaches at q ≈ 3.4 Å^-1, 3.6 Å^-1 and 1.6 Å^-1, respectively. The first crossing position of {\omega }_L and {\omega }_T for Ni₈₁B₁₉ metallic glass in the HB, TG and BS approaches is seen at q ≈ 3.4 Å^-1, 3.6 Å^-1 and 1.6 Å^-1, respectively. The first crossing position of {\omega }_L and {\omega }_T for Ni₃₃Y₆₇ metallic glass in the HB, TG and BS approaches is seen at q ≈ 1.3 Å^-1, 0.7 Å^-1 and 1.2 Å^-1, respectively. The first crossing position of {\omega }_L and {\omega }_T for Ni₃₆Zr₆₄ metallic glass in the HB, TG and BS approaches is seen at q ≈ 11.5 Å^-1, 1.5 Å^-1 and 1.1 Å^-1, respectively. The first crossing position of {\omega }_L and {\omega }_T for Ni₅₀Zr₅₀ metallic glass in the HB, TG and BS approaches is seen at q ≈ 1.5 Å^-1, 1.5 Å^-1 and 1.3 Å^-1, respectively. The first crossover position of {\omega }_L and {\omega }_T for Ni₆₀Nb₄₀ metallic glass in the HB, TG and BS approaches is observed at q ≈ 1.7 Å^-1, 1.6 Å^-1 and 1.1 Å^-1, respectively. The first crossing point between two phonon frequencies, {\omega }_L and {\omega }_T, in the HB, TG and BS approaches for Ni₈₁B₁₉ metallic glass is observed at q ≈ 2.9 Å^-1, 1.1 Å^-1 and 1.5 Å^-1, respectively. Moreover, the present outcome of PDC with BS approach is higher than those with HB and TG approaches. The present results of PDC for Ni₃₃Y₆₇ and Ni₅₀Zr₅₀ binary metallic glasses are found to be qualitatively in agreement with the available theoretical data [1,2].

The results for the PDC obtained from HB approach using the S-local field correction function are shown in Fig. 10. It is observed from Fig. 10 that, the first peak position of longitudinal branch of Ni₃₁Dy₆₉ glass is higher, while that of Ni₃₃Y₆₇ glass is lower in comparison with other metallic glasses. Also the first minima of longitudinal branch of Ni₈₁B₁₉ glass shows at higher q-values, while it shows for Ni₃₃Y₆₇ glass at lower q-values. The lower dip in the longitudinal branch justifies the correctness and stability of the pair potential. The same results are observed in transverse branch as well. Moreover, it is observed from Fig. 10 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes. This shows the existence of collective excitations at larger momentum transfer due to longitudinal phonons only and the instability of the transverse phonons due to the anharmonicity of the atomic vibrations in the metallic systems. Also in the high wave vector region, damping of phonons dominates the transverse mode, which is indicating the fluid characteristic of the glass, i.e., the transverse phonon behavior is monotonic. Here in transverse branch, the frequencies increase with the wave vector q and then saturate at ≈ q = 2.0 Å^-1, supporting the well known Thorpe model [35]that describes a glass like a solid containing finite liquid cluster. The transverse phonons are absorbed for frequencies larger than the smallest eigen frequencies of the largest cluster.

As shown in Figs. 11–16 for Ni₃₁Dy₆₉, Ni₃₃Y₆₇, Ni₃₆Zr₆₄, Ni₅₀Zr₅₀, Ni₆₀Nb₄₀ and Ni₈₁B₁₉ metallic glasses, the exchange and correlation functions also affected the anomalous behavior (i.e. deviation from the T^{\mathrm{3}} law), which is observed in the vibrational part of the low temperature specific heat C_V. The reason behind the anomalous behavior may be that the low frequency modes modify the generalized vibrational density of states of the glass with that of the polycrystal. These modes are mainly responsible for the difference in the temperature dependence of the vibrational part of the specific heat which departs from the normal behavior. The existence of a portion of the spectrum with ‘softer phonons’ (resembling rotons in liquid helium) may be the cause of anomalous behavior of low temperature specific heat C_V. In the low temperature region, a contribution to the low temperature specific heat C_V is made by phonons of the initial part of the frequency spectrum. When the temperature reaches a value at which the energy of the thermal motions becomes comparable to the energy of ‘softer phonons’ minimum, an additional contribution appears to heat capacity from the roton portion of the phonon frequency \omega (q). The present results of low temperature specific heat for Ni₆₀Nb₄₀ binary metallic glass are found to be qualitatively in agreement with the available theoretical data [36].

Furthermore, the thermodynamic and elastic properties of Ni-based amorphous alloys estimated from the elastic limit of the PDC are tabulated in Tables 2-7. The present results of Ni₃₃Y₆₇ glass are found lower in line with other theoretical data [1]. Also the present outcome of Ni₅₀Zr₅₀ glass shows lower results in comparison with the theoretical reported data [2]. It is seen that the screening theory plays an important role in the prediction of the thermodynamic and elastic properties of metallic glasses. The present values of the isothermal bulk modulus B_T change with the atomic volume of binary metallic glass, which shows close similarity in the response of crystalline and disordered structures to compressive stains. The present results of sound velocities of Ni₃₃Y₆₇ and Ni₅₀Zr₅₀ binary metallic glasses are more than five and nine times lower than the highest theoretical data [1,2]. The percentile influences of longitudinal sound velocity {\upsilon }_L of Ni₃₃Y₆₇ and Ni₅₀Zr₅₀ binary metallic glasses with respect to the available highest theoretical data [1,2] are found around 70.84%-85.71% and 80.46%-88.80%, respectively, while those influences of transverse sound velocity {\upsilon }_T are found around 76.18%-86.70% and 89.37%-93.91%. The percentile influences of isothermal bulk modulus B_T of Ni₃₃Y₆₇ and Ni₅₀Zr₅₀ binary metallic glasses with respect to the available highest theoretical data [1,2] are found around 90.45%-91.61% and 28.33%-76.63%, respectively. From Tables 2-7, it is noted that the thermodynamic and elastic properties of binary metallic glasses do not have much impact on considering screening effects. The present results computed from BS approach show higher values than those of the other two theoretical approaches. It is noted that, the inclusion of exchange and correlation effects with static H-dielectric function has enhanced the longitudinal and transverse sound velocities for HB and TG approaches, while for BS approach, suppression on both velocities is observed. No theoretical or practical data are available in the literature for any comparison of most of the binary metallic glasses, hence it is difficult to offer any remarks at this stage.

The quantitative difference between the present calculation and the experimental results, in spite of good qualitative agreement, can be attributed to the following conditions: i) the sampling conditions of the experiments, ii) the short supply of data in the long wavelength region and iii) the low or high effectiveness of the local field correction functions used for the calculation of the pair potential.

From the overall picture of the present study it is noticed that, the proposed model potential is successfully applicable to study the vibrational dynamics of Ni-based metallic glasses. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the binary metallic glasses are not available in the literature, but the present study is very useful in forming a set of theoretical data of particular metallic glass.

In all the three approaches, it is very difficult to judge which approach is the best for computations of phonon dynamics of Ni-based metallic glass, because each approximation has its own identity. The HB approach is the simplest and older one, which generates consistent results of the phonon data of these glasses, because the HB approach needs minimum number of parameters. Whereas, the TG approach is developed upon the quasi-crystalline approximation in which effective force constant depends upon the correlation function for the displacement of atoms, and correlation function of displacement itself depends on the phonon frequencies. The BS approach has retained the interatomic interactions effective between the first nearest neighbors only, hence, the disorderness of the atoms in the formation of metallic glasses is more, which shows deviation in magnitude of the PDC and their related properties. From the present study we have concluded that all the three approaches are suitable for studying the phonon dynamics of the amorphous materials. In that sense, successful application of the model potential with the three approaches is observed from the present study.

Conclusions

Last, it is concluded that, in the study of phonon dynamics of metallic glasses, the pair potential and its derivatives as well as pair correlation function play an important role. In the present computation, the WH form is adopted to generate the pair potentials, which ignores the angular interaction due to partially filled d-bands in transition metals. Most recent model potential with WH model and HB approach produces consistent results of phonon dynamics for all metallic glasses. That said, the present model potential is suitable for studying the phonon dynamics of six Ni-based metallic glasses, which confirms the applicability of the model potential in the aforementioned study. Such study on phonon dynamics of other binary as well as ternary liquid alloys and metallic glasses is in progress, which will be communicated in near future.

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