BIBECHANA 18 (1) (2021) 33-47 

33 
 

BIBECHANA 
ISSN 2091-0762 (Print), 2382-5340 (Online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal 

 

 

Phonon dynamics of Zr67Ni33 and Fe80B20 binary glassy alloys 
 

 

Aditya M. Vora1*,  Alkesh L. Gandhi2**  
1Department of Physics, University School of Sciences, Gujarat University, Ahmedabad, 380 009, 

Gujarat, India. 
2Department of Physics, B. V. Shah (Vadi Vihar) Science College, C. U. Shah University, Wadhwan, 363   

 002, Gujarat, India. 
*Email: amvora@gujaratuniversity.ac.in,  voraam@gmail.com   
**Email: gandhi.alkesh.jain@gmail.com 
 

 

Article Information: 

Received: May 4, 2020 

Accepted: June 17, 2020 

 

Keywords:  

Bulk metallic glass  

Model potential  

Interatomic pair potential 

Phonon dispersion curve (PDC) 

Thermodynamic properties  

 

 

 

 

ABSTRACT 

Binary amorphous alloys are the primary bulk metallic glasses (BMGs). Two 

binary BMGs Zr67Ni33 and Fe80B20 have been studied in the present work using 

the pseudo- alloy-atom (PAA) model based on the pseudopotential theory. Some 

important thermodynamic properties like Debye temperature and elastic 

properties like elasticity moduli and Poisson’s ratio at room temperature are 

theoretically computed with the help of pseudopotential theory from the elastic 

limit of the phonon dispersion curves (PDCs). The collective dynamics of 

longitudinal and transverse phonon modes are investigated in terms of 

eigenfrequencies of the localized collective modes. The presently computed 

results are compared with the other such data including theoretically generated 

results from the molecular dynamics at different temperatures as available in the 

literature and an acceptable agreement is found.   

DOI: https://doi.org/10.3126/bibechana.v18i1.28760 
This work is licensed under the Creative Commons CC  BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 
 

 
1. Introduction 

 

Generally, the bulk metallic glasses (BMGs), which 

are non-crystalline metallic alloys, can be made-up 

with low cooling rates in many alloy systems. A 

metallic material is formed by the basic units of 

well-arranged metallic microstructures designed by 

the single-crystal grains of varying sizes. An alloy 

is produced by melting the constituent metal 

elements in predefined proportions followed by the 

solidification process. The solidification process of 

an amorphous alloy has a deep impact on its 

properties and structure at the corresponding 

temperature. So, the data of the physical properties 

of the molten alloys, before its solidification, is 

necessary for the development of a BMG with 

predefined characteristics. The properties of an 

alloy are fully dependent on the crystallization 

procedure and the final metallic microstructure, 

where both are determined by the selection of 

cooling pattern and the adopted thermal treatments. 

Unlike conventional crystalline alloys, a BMGs is 

http://nepjol.info/index.php/BIBECHANA
mailto:amvora@gujaratuniversity.ac.in
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mailto:gandhi.alkesh.jain@gmail.com
https://doi.org/10.3126/bibechana.v18i1.28760
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Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

34 
 

produced by quenching the alloy melt below its 

crystalline temperature by preventing the regular 

long-range arrangement with crystallization.  The 

quickly frozen alloy-melt keeps the precursor liquid 

structure. Thus, the BMGs are said to be solids with 

liquid-like non-crystalline structure at the atomic or 

molecular level, which makes them highly resistant 

to permanent distortion and makes them more than 

twice tougher to its crystalline counterpart. In 

recent decades, the neutron scattering data from 

short-wavelength phonons in many BMG-

configurations have been increasingly available. 

There are found an interesting similarity between 

the scattered neutron spectra of liquids and that of 

amorphous or crystalline solids near their melting 

point. Further, the theory of lattice dynamics is 

equally and successfully applied to the ordered 

crystalline as well as amorphous materials [1-32]  
 

1.1 𝐙𝐫𝟔𝟕𝐍𝐢𝟑𝟑 BMG 
 

In the binary transition metal-metal alloy systems 

i.e. Zr-Ni BMG is a unique one and its 

crystallization temperature is higher than the glass 

transition temperature (652K) under the heating 

conditions. Suck et al. [33] have studied 

longitudinal phonon dispersion mode of Zr67Ni33 

glass by neutron inelastic scattering (NIS) method. 

While, Gupta et al. [13]  have theoretically studied 

the phonon dispersion curves (PDCs) of such glass 

using Hubbard-Beeby (HB) [30] and Bhatia-Singh 

(BS) ([28] computational approaches and compared 

their outcomes with the results achieved through 

molecular dynamics (MD) simulation by Aihara et 

al. [15,16] . Also, they have found that there is a 

good agreement between the experimentally 

observed and theoretically computed data. Otomo 

et al. [17] have studied the dynamical structure 

factors and collective excitations of 𝑍𝑟67𝑁𝑖33 glass 

with NIS method. Similarly, Lad and Pratap [18] 

have theoretically investigated the PDC of Zr-Ni 

BMG with model potential formalism using TG 

[31,32] and BS [28] approaches and got the results 

very close to the experimental and theoretical data 

available in the literature. While, Vora and co-

workers [1-12] have evaluated the collective 

dynamics of some BMGs using Ashcroft’s empty 

core model pseudopotential [34]  Singh et al. [19] 

have examined Zr-Ni BMG with HB [30], TG 

[31,32] and BS [28] approaches using a newly 

constructed free model potential, which is, in fact, 

the alteration of the Ashcroft's model potential [34] 

only. The electronic structure, bonding and 3D 

atomic structure of Zr67Ni33 BMG were studied by 

Sugita et al. [20]. Very recently, Gandhi and Vora 

[6-10] have reported vibrational dynamics of some 

binary to hexanary BMGs using model 

pseudopotential theory. The interest in the Zr-based 

BMGs has been growing due to their superior 

mechanical properties like high fracture resistance, 

good corrosion resistance and micro-formability, 

which make them a favorable choice for various 

applications including high field superconducting 

magnets and the place where brittle alloys are not 

used [19,20]. 
 

1.2 Fe80B20 BMG 
 

The metal-metalloid type Fe80B20 BMG, mainly 

referred to be as Metglas-2605, is produced by the 

rapid solidification process. It is a ferromagnetic 

with the good soft magnetic property having a high 

yield strength of ~3630 MPa, high elastic stiffness, 

and thermal stability, which all together make it 

superior to many other commercial binary BMGs. 

Though a little plasticity is its minor drawback, the 

low-cost Fe80B20 BMG is preferable for structural 

applications [21]. In the early 80's, Devis et al. [35] 

have experimentally studied properties of Fe80B20 

BMG. Dimitrov et al. [22] and Agarwal et al. [23, 

24] have carried out a limited study of this BMG. 

While, Vora [3] has theoretically studied Fe80B20 

using all three theoretical approaches of PDCs. 
 

Considering the multilevel applications of the 

BMGs, the vibrational dynamics of two binary 

BMGs Zr67Ni33 and Fe80B20 are reported in the 

current work. The well-behaved Shaw’s optimized 

constant-core model pseudopotential [36] is 

employed using a self-consistent phonon scheme 

by relating multiple scattering and phonon 

eigenfrequencies. Such phonon frequencies 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

35 
 

represent many-body correlation functions of atoms 

and concerned interatomic potential. To study the 

screening dependency on the aforesaid properties, 

the most effectively used five different local field 

correction (correlation) functions namely Hartree 

(H) [37], Taylor (T) [38] Ichimaru-Utsumi (IU) 

[39] and Sarkar et al. (S) [40] have been adopted to 

determine the effect of exchange and correlation on 

the bare ion potential. These screening functions 

help us to investigate the screening influence on the 

vibrational dynamics of the selected binary BMGs. 

Three independent computational approaches viz. 

Hubbard-Beeby (HB) [30], Takeno-Goda (TG) [31, 

32], Bhatia-Singh (BS) [28] and Shukla and 

Campnha [29] have been employed to generate 

PDCs of said binary BMGs. The important 

thermodynamic and elastic properties viz. 

longitudinal sound velocity (𝜐𝐿), transverse sound 

velocity (𝜐𝑇), Young’s modulus (𝑌), isothermal 

bulk modulus (𝐵𝑇), rigidity modulus (K), 

Poisson’s ratio (𝜎) and Debye temperature 

(𝜃𝐷) have been computed using the long 

wavelength limits of PDCs. 
 

2. Computational methodology 
 

 

The pseudopotential perturbation theory is an 

accepted tool for determining properties of 

amorphous alloys. As far as analytical calculations 

are concerned, many approximations based on 

model approaches are available to study the BMGs 

[1-11]. Shaw’s optimized constant-core model 

pseudopotential [36] along with five different local 

field correction functions viz. H, T, IU, F and S 

have used for studying the screening influence on 

the said properties of the selected BMGs.   
 

Here, the three theoretical approaches, namely HB 

[30], TG [31,32] and BS [28] have been 

extensively employed to study the phonon 

dispersion and collective excitations in binary 

BMGs. However, Hubbard and Beeby (HB) [30] 

have derived expressions for the longitudinal and 

transverse phonon eigenfrequencies by examining 

collective motion in the amorphous liquid system 

as a generalization of the phonon theory of solids 

using a random phase approximation. In which, the 

product of the static pair correlation function and 

the second order derivative of the interatomic 

potential is peaked near the hard-core radius. 

While, Takeno and Goda (TG) [31, 32] have 

expressed longitudinal and transverse phonon 

eigenfrequencies in terms of many-body correlation 

functions of atoms and interatomic potentials in 

amorphous solids. Bhatia and Singh (BS) [28, 29] 

have proposed a phenomenological model to 

determine the longitudinal and transverse phonon 

eigenfrequencies in amorphous solids.  
  

In most of such studies, the pseudopotential 

parameters are evaluated in such a way that it can 

generate the comparable results of the pair 

correlation function (PCF) (g(r)) and PDCs, which 

are found to be in good agreement with 

experimental data. Mostly, Vegard’s law is used to 

explain electron-ion interactions for binary BMGs. 

However, the well suitable pseudo-alloy-atom 

(PAA) model explains such interactions vary 

precisely. Hence, in the present article, the PAA 

model is used to investigate the phonon dynamics 

of 𝐴𝑥𝐵100−𝑥 binary BMG systems [1-11]. 
 

Considering these binary combinations of 

amorphous alloys to be mono-component fluids, 

the effective ion-ion interactions are given by [1-

11], 

𝑉 (𝑟) =
𝑍2𝑒2

𝑟
+

2

𝜋
 ∫ 𝐹(𝑞) 𝑒−𝑖𝑞𝑟 𝑑𝑞                      (1) 

 

where, 𝐹(𝑞) is the characteristics wave number, 𝑞 

is the space vector. The first and second terms in 

the above equation represent Coulombic 

interactions between ions and indirect interactions 

between conduction electrons, respectively. The 

pseudopotential perturbation along with the linear 

screening theory provides the basis of interatomic 

pair potential. The free-electron theory of metals 

has been extended to introduce the effects of 

amorphous structure. The presently computed 

interatomic pair potential (V(r)), based on the 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

36 
 

extended theory of Wills-Harrison model [41] for the BMGs is given by, 

𝑉(𝑟)= 𝑉𝑠(𝑟) + 𝑉𝑏(𝑟) + 𝑉𝑟(𝑟) 

𝑉(𝑟) =  {
𝑍𝑠
2𝑒2

𝑟
+ 

𝛺0

𝜋2
 ∫ 𝐹(𝑞) [

sin(𝑞𝑟)

(𝑞𝑟)
] 𝑞2 dq } + {− 𝑍𝑑 [1 −

𝑍𝑑

10
] (

12

𝑁𝑐
)

1

2
(
28.6

𝜋
) (

2 𝑟𝑑
3

𝑟5
)} + {𝑍𝑑 (

450

π2
)
2
(
𝑟𝑑
6

𝑟8
) }      (2)   

where 𝑉𝑠(𝑟) is the 𝑠-electron contribution. The term 

𝑉𝑏(𝑟) considers the Friedel-model band broadening 

contribution to the transition metal cohesion and 

𝑉𝑟(𝑟) came from the repulsion of the d-electron 

muffin-tin orbital on different sites owing to their 

non-orthogonality. Also, the Z is the valence and  

𝛺0 the atomic volume of the BMGs, respectively. 

The self-consistent band structure calculation can 

provide the partials-density of states which can be 

used to obtain 𝑍𝑠. The characteristics energy wave 

number term 𝐹(𝑞), with 𝑞 as the 𝑞-space vector, 

under integration in the above equations, can be 

written as [1-11], 
 

𝐹(𝑞) =  
−𝛺0𝑞

2

16𝜋
 |𝑊𝐵(𝑞)|

2  
[ 𝐻(𝑞)−1]

{1+[ 𝐻(𝑞)−1][1−𝑓(𝑞)] }
, (3)        

 

where 𝑊𝐵(𝑞), 휀𝐻(𝑞) and 𝑓(𝑞) are the bare ion 

potential, the Hartree dielectric response screening 

function [37] and the local field correction 

functions needed to introduce the exchange and 

correlation effects, respectively. Shaw’s model 

potential used in the present work is of the form 

[36] 

𝑊𝐵(𝑞) = −
8𝜋𝑍

𝛺0𝑞
2 [
sin𝑞𝑟𝑐

𝑞𝑟𝑐
]                           (4) 

 

where, the potential parameter is computed by 𝑟𝑐 =

 (0.51) 𝑟𝑆 Z
−1/3 [1-2].  

Here, Wigner-Seitz radius of the BMG is given by 
 

 𝑟𝑠 = (
3𝛺0

4𝜋𝑍⁄ )
1/3

             
 

For BMGs, the PCF (g(r)) discloses much useful 

information about the inter-particle radial 

correlation, structure and further, that predicts the 

electrical, elastic, thermodynamic and other 

physical properties. Either it is obtained by using 

X-ray diffraction or by neutron scattering 

experimentally [25-27,42] theoretically [1-12, 23-

24, 43] from interatomic pair potentials. 
 

The vibrational dynamics of the aforementioned 

binary BMGs are studied by computing the 

longitudinal and transverse phonon 

eigenfrequencies, where the derivative of the 

interatomic pair potential (V(r)) in combination 

with PCF (g(r)) are used to compute the PDCs. 

The product of the static pair correlation function 

(g(r))  and the second order derivative (V′′(r)) of 

the interatomic potential is peaked at the hard-

sphere diameter. The rapid atomic motions of a 

disordered amorphous systems are very important 

because their structural, thermodynamic and 

transport properties are described by collective 

atomic motions. 
 

According to HB [30]-approach, the expressions 

for the two components of the phonon frequencies 

i.e. longitudinal and transverse are computed from 

[30], 

𝜔𝐿
2(𝑞) =  𝜔𝐸

2  [1 − 
sin(𝑞𝜎)

𝑞𝜎
− 6 

cos(𝑞𝜎)

(𝑞𝜎)2
+ 6 

sin(𝑞𝜎)

(𝑞𝜎)3
]    (5)  

                            

𝜔𝑇
2(𝑞) =  𝜔𝐸

2  [1 + 3 
cos(𝑞𝜎)

(𝑞𝜎)2
− 3 

sin(𝑞𝜎)

(𝑞𝜎)3
]   (6) 

with the maximum phonon frequency, 
 

𝜔𝐸
2(𝑞) =  (

4𝜋𝜌

3𝑀
) ∫ 𝑔(𝑟)𝑉′′(𝑟)𝑟2𝑑𝑟

∞

0
    (7) 

<      
Where 𝑀 is the atomic mass, 𝜌 the number density 

and 𝑉′′(𝑟)  the second derivative of the interatomic 

pair potential, respectively. 
 

While, Takeno and Goda [31, 32] have derived the 

following expressions for wave vector (q) 

dependent longitudinal and transverse phonon 

eigenfrequencies as follows, 

𝜔𝐿
2(𝑞) =

 (
4𝜋𝜌

𝑀
) ∫ 𝑑𝑟 𝑔(𝑟)

{
 
 

 
 [𝑟 𝑉′(𝑟) (1 −

sin(𝑞𝑟)

𝑞𝑟
)]

+ [𝑟2𝑉′′(𝑟) − 𝑟𝑉′(𝑟)] 

(
1

3
−

sin(𝑞𝑟)

𝑞𝑟
− 2

cos(𝑞𝑟)

(𝑞𝑟)2
+ 2

sin(𝑞𝑟)

(𝑞𝑟)3
)
}
 
 

 
 

∞

0

                   (8) 

and, 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

37 
 

𝜔𝑇
2(𝑞) =(

4𝜋𝜌

𝑀
)∫ 𝑑𝑟 𝑔(𝑟) {

[𝑟 𝑉′(𝑟) (1 −
sin(𝑞𝑟)

𝑞𝑟
)] +

 [𝑟2𝑉′′(𝑟) − 𝑟𝑉′(𝑟)] (
1

3
+ 2

cos(𝑞𝑟)

(𝑞𝑟)2
+ 2

sin(𝑞𝑟)

(𝑞𝑟)3
)
}

∞

0
            (9) 

 

According to improved BS model, the phonon 

frequencies of two branches are represented by [28, 

29]  

𝜔𝐿
2(𝑞) =

2𝑁𝐶

𝜌𝑞2
(𝛽𝐼0 + 𝛿𝐼2) +

𝑘𝑒𝑘𝑇𝐹
2 𝑞2|𝐺(𝑞𝑟𝑆)|

2

𝑞2+𝑘𝑇𝐹
2 (𝑞)

      (10) 

and, 

𝜔𝑇
2(𝑞) =

2𝑁𝐶

𝜌𝑞2
(𝛽𝐼0 +

1

2
𝛿(𝐼0 − 𝐼2))            (11) 

with,  

 

𝛽 =
𝜌𝑎2

2𝑀
[
1

𝑟

𝑑𝑉(𝑟)

𝑑𝑟
]
𝑟=𝑎

                (12)

       

𝛿 =
𝜌𝑎3

2𝑀
[
𝑑

𝑑𝑟
(
1

𝑟

𝑑𝑉(𝑟)

𝑑𝑟
)]
𝑟=𝑎

                          (13)

  

The notations of 𝐼0and 𝐼2 are, with 𝑥 = 𝑞𝑎, 

𝐼0 = 1 −
𝑠𝑖𝑛(𝑥)

𝑥
                            (14) 

and, 

𝐼2 =
1

3
− 𝑠𝑖𝑛(𝑥) [

1

𝑥
−

2

𝑥3
] −

2 𝑐𝑜𝑠(𝑥)

𝑥2
            (15)       

 

Here, 𝑀, 𝜌 and 𝑁𝐶  are the atomic mass, the number 

density and the coordination number of the said 

BMGs while 𝑉′(𝑟) and 𝑉″(𝑟) be the first and 

second derivatives of the interatomic pair potential, 

respectively.  
  

For the long-wavelength limit of the phonon 

frequency spectrum, the longitudinal and transverse 

frequencies are proportional to the wave vectors as 

follows [1-14, 18, 19, 23, 24, 28-32] . 

  𝜔𝐿 = 𝐿 𝑞   and     𝜔𝑇  =  𝑇 𝑞                        (16)                                                                                                       
 

For HB [30], TG [31, 32] and BS [28] approaches 

respectively, the formulations of longitudinal sound 

velocity(𝑣𝐿), and transverse sound velocity(𝑣𝑇 )  

are given by the following equations, 

𝑣𝐿  (𝐻𝐵) =   𝜔𝐸 (
3

10
𝜎2)

1/2
= (0.5477)𝜔𝐸 𝜎   (17)                                                                 

𝑣𝑇 (𝐻𝐵) =   𝜔𝐸 (
1

10
𝜎2)

1/2
= (0.3162)𝜔𝐸 𝜎   (18)                                                              

𝑣𝐿 (𝑇𝐺) =  {(
4𝜋𝜌

30𝑀
) ∫ 𝑑𝑟 𝑔(𝑟) 𝑟3

∞

0
[𝑟 𝑉′′(𝑟) − 4𝑉′(𝑟)]}

1

2
  (19)      

𝑣𝑇(𝑇𝐺) =  {(
4𝜋𝜌

30𝑀
) ∫ 𝑑𝑟 𝑔(𝑟) 𝑟3

∞

0
[3𝑟 𝑉′ (𝑟) − 4𝑉′(𝑟)]}

1

2(20)  

𝑣𝐿 (𝐵𝑆) =  {(
𝑁𝑐

𝜌𝑎
) [

𝛽

3
+
𝛿

5
 ] +

𝑘𝑒

3
}

1

2
             (21)                                                                                         

𝑣𝑇(𝐵𝑆) =  {(
𝑁𝑐

𝜌𝑎
) [

𝛽

3
+

𝛿

15
 ]}

1

2
                              (22)                                                                                               

 
 

For the selected BMGs, some of the important 

properties viz. isothermal bulk modulus (𝐵𝑇), 

rigidity modulus (G), Poisson’s ratio (σ ), Young’s 

modulus (𝑌) and Debye temperature 𝜃𝐷  are 

computed using the expressions as given below [1-

14, 18, 19, 23, 24,  28-32]. 

Isothermal bulk modulus 

 𝐵𝑇 = 𝜌𝑀 (𝑣𝐿
2 −

4

3
𝑣𝑇
2)                 (23)  

 

Modulus of rigidity 
   

𝐺 = 𝜌𝑀𝑣𝑇
2                 (24)                                                                               

Poisson’s ratio  𝜎 =
1

2
[
1−2 (

𝑣𝑇
2

𝑣𝐿
2⁄ )

1− (
𝑣𝑇
2

𝑣𝐿
2⁄ )

]                   (25)  

 

Young’s modulus  
  

𝑌 = 2𝐺(𝜎 + 1)               (26)                                                                   

Debye temperature  𝜃𝐷 =
ħ 𝜔𝐷

𝑘𝐵
 =

ħ 

𝐾𝐵
  2𝜋 (

9𝜌

4𝜋
)

1

3
(
1

𝑣𝐿
3 + 

2

𝑣𝑇
3)
−
1

3
                        (27)                                           

 

 

The input parameters used for present 

computational work are tabulated in Table 1, which 

are calculated from using PAA model [1-11] by 

adopting pure standard data of constituent element 

of BMGs [41]. 

 

 

 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

38 
 

Table 1: Input parameters and constants used in present work. 
 
 

BMG 𝒁 𝒁𝑺 𝒁𝒅 𝑹𝒅(𝐚𝐮) 𝑹𝒔(𝐚𝐮) 𝜴𝟎(𝐚𝐮)
𝟑 𝑵𝑪 M (amu) 𝒓𝒄 (𝐚𝐮) 

Zr67Ni33  3.00 1.50 5.20 1.21 1.96 95.02 6.40 46.84 0.6941 

Fe80B20 3.34 1.50 4.48 2.23 2.10 129.48 12.00 80.49 0.7163 
 

 

 

3. Results and Discussion 
 
 

The discussion and comparison of our present 

results with available outcomes either theoretical or 

experimental in the literature are narrated in this 

section briefly.  
  

3.1 Interatomic pair potential 
 

The pair potentials 𝑉(𝑟) (in Ryd.) versus 

interatomic distance 𝑟(au) are displayed in Figs. 

1(a) and 1(b) of both BMGs and compared with 

available theoretical data [12-14]. Also, some 

theoretical observations from interatomic pair 

potential curves are presented in Tables 2-3. 

Though the shape and nature of the interatomic pair 

potentials appear similar type while the potential 

well depths related to different BMGs are appeared 

differently. It is observed that the location of the 

well-depth of computed pair potentials 𝑉(𝑟) moves 

towards the left for both BMGs as compared to that 

of reported theoretical data [12-14], which show 

higher results than the computed one.   
 

The first minima position is significantly affected 

by the types of screening i.e. the similarity is seen 

in the pair potential nature by changing the local 

field correction functions except at or near the dip 

position. Under the exchange and correlation effect, 

inclusion in the static H-dielectric screening 

changes the width and depth of the interatomic pair 

potential curves. The well-depth impact due to H-

correction function is slightly less than that due to 

the rest of the screening functions. The first zero in 

the pair potential arises at  𝑟 =  𝑟0 = 1.94 au and 

3.76 au for the  𝐹𝑒80𝐵20  and  𝑍𝑟67𝑁𝑖33 BMGs 

respectively. Thus, the effect of exchange and 

correlation function is not found significantly, but 

the potential  well  is  more in case of other   

 

 
correlation functions than that of the H-function. 

Various screening functions influence to increase 

the well-depth slightly. The maximum and 

minimum well-depth for 𝐹𝑒80𝐵20 is seen due to S- 

and T-functions, respectively, while the same for 

the rest of three are lying between them. Similarly, 

in the case of 𝑍𝑟67𝑁𝑖33, the maximum and 

minimum well-depths are found for IU- and H-

screening functions and the rest of three are lying 

between them. The interatomic pair potential well-

depth deepens with increasing atomic volume and 

hence here, well-depth of 𝐹𝑒80𝐵20 is more than that 

of 𝑍𝑟67𝑁𝑖33. The potential energy remains negative 

within the large 𝑟-region. The flat type negligible 

and gradually vanishing oscillations after 𝑟 =7.31 

au and 𝑟 = 9.61 au in  𝐹𝑒80𝐵20  and 

 𝑍𝑟67𝑁𝑖33 BMGs are observed due to the 

interaction between electrons and ions because of  

Colombian repulsion of the potential. However, we 

have not shown these features correctly from the 

graph. For the small values of interatomic distances 

𝑟, the interatomic pair potentials 𝑉(𝑟) are strongly 

repulsive and with the increase of distances, less 

oscillating nature are observed as expected, which 

is directly related to the Friedel oscillations due to 

log-singularity in electron susceptibility function. 

Moreover, the interatomic pair potentials after the 

first minima or dip, show minor oscillations of very 

small amplitude and converges to zero at some 

larger interatomic distances. In the attractive part, 

the interatomic pair potentials converge to a limited 

and fixed value rather than being zero. While, 

Bretonnet and Derouiche [44] showed that the 

repulsive part of 𝑉(𝑟) is drawn lower and its 

appealing part is deeper because of the 𝑑-electron 

impact where the 𝑉(𝑟) is moved towards the lower 

𝑟-values and supports the 𝑑-electron effect. All pair-potential 

graphs show the combined effect of 𝑠-and 𝑑-electrons.  

 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

39 
 

Table 2 :  Observations of Interatomic Pair Potentials for Fe80B20 BMG. 
 

Screening Function H T IU F S 

First Zero r (au) 1.94 1.94 1.94 1.94 1.94 

Maximum Depth r (au)  2.27 2.25 2.23 2.25 2.25 

Maximum Dip (Ryd.) −0.56 −0.56 −0.57 −0.57 −0.62 

Oscillatory Behaviour 

starts from r (au) 

7.92 6.18 6.34 6.07 6.75 

Table 3 :  Observations of Interatomic Pair Potentials for Zr67Ni33 BMG. 
 

Screening Function H T IU F S 

First Zero r (au) 3.76 3.76 3.76 3.76 3.76 

Maximum Depth r (au)  4.44 4.46 4.52 4.51 4.41 

Maximum Dip (Ryd.) −0.10 −0.10 −0.12 −0.11 −0.11 

Oscillatory Behaviour 

starts from r (au) 

10.50 12.54 10.43 9.43 10.10 

 
3.2 Phonon dispersion curves 
 

Figures 2-4 show the PDCs of the both BMGs 

under investigation generated from the interatomic 

pair potential and pair correlation functions using 

HB [30]-, TG[31,32]- and BS [28]- approaches 

with the five screening functions viz. H, T, IU, F 

and S, respectively. In these PDCs, the longitudinal 

and transverse modes of phonon eigenfrequencies 

have been plotted in the form of 𝜔𝐿 and 𝜔𝑇 (×

1013𝑠−1) versus 𝑞 (Å−1) graphs. Table 4 shows a 

comparative observational analysis from the PDCs 

of both BMGs.  The presently computed outcomes 

are compared with available either theoretical [14-

16] or experimental data considered by Gupta et al. 

[13]. The presently obtained results of PDCs are 

found qualitative agreement with them. Also, the 

MD result of the Aihara et al. [15,16] of 

 𝑍𝑟67𝑁𝑖33 BMG at two different temperatures 100K  

 

 

 
 

and 500K indicates the low and high amorphous 

states respectively, which are also shown in the 

same figure. 
 

For  𝐹𝑒80𝐵20  BMG, the first and second 𝜔𝐿- 

𝜔𝑇 crossover positions for the HB [30], TG [31,32]  

and BS [28] approaches are seen at 2.5 (Å−1) and 

3.9; 1.3 (Å−1) and 2.2 (Å−1); 2.3 (Å−1) and 2.6 

(Å−1), respectively. For  𝑍𝑟67𝑁𝑖33 BMG, the first 

and second 𝜔𝐿- 𝜔𝑇 crossover positions for the HB 

[30], TG [31,32]  and BS [28] approaches are seen 

at 2.3 (Å−1) and 3.6(Å−1); 1.3(Å−1) and 

2.1(Å−1); 2.2(Å−1) and 2.6 (Å−1), respectively. 

Actually, here model potential parameter 𝑟𝐶 is 

calculated from the well-known formula, which 

shows significant difference than the reported data. 

Moreover, the present outcomes of PDCs due to 

TG [31,32] approach is higher  than those due 

to HB [30] and BS [28] approaches. From 

the PDCs, the frequency increases with the wave  

 

 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

40 
 

 

 

 

(a)        (b) 

Fig. 1: Pair potentials for (a) Fe80B20 and (b) Zr67Ni33 BMGs. 

 

(a)                                                                             (b)

Fig, 2: Phonon dispersion curves for (a) Fe80B20 and (b) Zr67Ni33 BMGs using HB approach. 

         



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

41 
 

  

(a) 

 

    (b) 

Fig. 3: Phonon dispersion curves for (a) Fe80B20 and (b) Zr67Ni33 BMGs using TG approach. 

 

(a)                                                   (b) 

Fig. 4: Phonon dispersion curves for (a) Fe80B20 and (b) Zr67Ni33 BMGs using BS approach. 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

42 
 

  

Table 4 : Comparison of the PDCs for the selected binary BMGs. 
 

BMG Freq. Order Maxima  𝒒(Å−𝟏) Minima  𝒒(Å−𝟏) 

[30] [31,32] [28] [30] [31,32] [28] 

F
e 8

0
B

2
0
 

𝝎𝑳 

 

First  1.7 0.7 0.9 3.2 1.7 2.4 

Second 4.7 2.5 3.0 --- 3.3 3.7 

Third --- 4.0 4.2 --- 4.7 4.7 

𝝎𝑻 First  2.5 1.3 0.9 4 2.1 2.6 

Second --- 2.9 3.1 --- 3.6 3.7 

Third --- 4.3 4.2 --- --- 4.7 

Z
r 6

7
N

i 3
3
 

𝝎𝑳 

 

First  1.5 0.7 0.8 2.9 1.6 2.4 

Second 4.2 2.4 2.9 4.8 3.1 3.5 

Third --- 3.7 4.0 --- 4.4 4.6 

𝝎𝑻 First  2.3 1.2 0.9 3.6 2.0 2.5 

Second 4.8 2.6 3.0 --- 3.2 3.5 

Third --- 3.9 4.1 --- 4.5 4.6 

 
vector and then saturates after a certain level 

suggesting that the selected BMGs contain finite 

liquid clusters, which confirms the Thorpe theory 

[45]. The transverse phonons are absorbed for such 

frequencies as they are larger than the smallest 

eigenfrequencies of the largest cluster. 
 

These PDCs reveal that the oscillations are more 

prominent in longitudinal phonon modes than those 

in the transverse phonon modes in the large wave 

vector transfer region, which corresponds to the 

fluctuations in the particle density of these BMGs 

and therefore, approves the presence of the short-

wavelength collective excitations. The instability in 

the transverse mode oscillations is due to the 

anharmonicity of the atomic vibrations, which 

suggests the transverse phonons undergo a large 

thermal modulation. The fluctuations in the 

transverse phonon mode decrease out rapidly, 

which connects to the damping transverse phonon 

modes in BMGs. In the lower wave vector transfer  

region, the PDCs are linear due to the 

characteristics of the elastic waves; but for the 

 
 

higher wave vector region, this linear relationship  
 

does not hold, as the sound velocity is different for 

different wave vectors. The collective excitations at  

larger wave vector transfer are due to the eminence 

of the longitudinal mode excitations only. The 

longitudinal part of PDC shows that the BMGs are 

screening sensitive in the low momentum region. 

The difference in the 𝜔𝐿 or 𝜔𝑇 (𝑠−1) versus 𝑞 

(Å−1) exists from the beginning, becomes 

maximum at the peak point and again decreases till 

the dip point. The peak height is dependent on the 

screening function as well as on the applied 

pseudopotential approach. The longitudinal PDCs 

display oscillatory behavior for larger 𝑞 values, 

which is absent in case of transverse phonons. A 

longitudinal phonon curve reaches maxima at a 

higher 𝑞 value than the transverse one. 

Comparatively, the longitudinal frequency bears 

higher numerical value than the transverse 

frequency for a particular 𝑞 value, which is 

reflected by the distance between their peak or deep 

positions. For all three approaches, the transverse 

phonon frequencies grow with wave number and 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

43 
 

become saturated at the first peak with small 

variations. Though the first peak position is 

independent of the screening functions, the peak-

height is fairly dependent on the screening function 

used in this computational work. From the second 

and third ordered minima and maxima for any 

screening functions in the HB [30]-approach do not 

appear clearly. The selected BMGs are screening 

sensitive in the low-momentum region. 

Furthermore, the phonons are detected as sound 

waves in the macroscopic frequency limits of the 

spectrum. Hence, the longitudinal sound velocity 

(𝑣𝐿) and transverse sound velocity (𝑣𝑇) have been 

estimated from the linear part of their curves.  
  

3.3 Computed properties 
 

The computed elastic and thermodynamic 

properties of Fe80B20  and Zr67Ni33 BMGs with the 

available reference data are presented by Tables 5  

and 6 respectively, from which, it appears that the 

calculated values of different properties nearly 

agree to each other for different screening functions  

for a particular approach but considerably differs 

for different approaches i.e. a property holds nearly 

the same value for all screening functions for a 

particular approach, but the same property holds  

different values for the same screening function in 

another approach. For the selected binary glasses, 

our presently computed outcomes from HB and BS 

approaches are found in fair qualitative agreement 

with available results [2, 5, 13-16, 19, 21]. Our 

results are over estimated with them. 
 

It is quite tough to judge that which of the three 

approaches adopted here is the best suitable for  

computation of vibrational dynamics of aforesaid  

 

binary BMGs, because each approach has its own  

identity, importance and limitations too. The HB 

approach is an older one but simple enough to need 

a minimum number of parameters to produce 

consistent results of the phonon data. While, the 

TG [31,32] approach is established upon the quasi-

crystalline approximation, where effective force 

constant depends upon the correlation function for 

the displacement of atoms, which further depends 

on the phonon frequencies. Similarly, the BS 

approach retains interatomic interactions 

effectively between the first nearest neighbours 

only. Therefore, the atomic disorderedness in the 

formation of metallic glasses is more which show a 

deviation in the magnitude of PDCs and their 

concerned properties.  
 

The dielectric function plays an significant role in 

the estimation of potential due to the screening of 

the electron gas. For this determination, in the 

present study, the local filed correction function 

due to H, T, IU, F and S are taken.  Motive for 

choosing them is that, the H-function does not 

include exchange and correlation effect and 

signifies only static dielectric function, whereas the 

T-function cover the overall features of the 

numerous local field correction functions projected 

before 1972. The IU, F and S functions are newer 

one amongst the existing functions and not 

exploited thoroughly in current study. This helps us 

to study the comparative effects of exchange and 

correlation in the aforementioned properties. 

Therefore, the present outcomes found from five 

dissimilar local field correction functions may be as 

in contract with each other, as different order of 

magnitude in the Figures 1-4.       

 

 

 

 

 
 

 

 

 

 

 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

44 
 

  

Table 5: Thermodynamic and elastic properties of Fe80B20 BMG. 

 

 

 

 

 

 

 

 

A
p

p
ro

a
ch

 

S
cr

. 
F

u
n

. 𝐯𝐋 × 𝟏𝟎
𝟓 

(𝐜𝐦/𝐬) 

𝐯𝐓 × 𝟏𝟎
𝟓 

(𝐜𝐦/𝐬) 

𝐁𝐓 × 𝟏𝟎
𝟏𝟏 

𝐝(𝐲𝐧𝐞/𝒔𝟐) 

𝐆 × 𝟏𝟎𝟏𝟏 

(𝐝𝐲𝐧𝐞/𝒔𝟐) 

σ 𝐘 × 𝟏𝟎𝟏𝟏 

(𝐝𝐲𝐧𝐞

/𝒔𝟐) 

𝛉𝐃   K 

[3
0

] 

H 8.23 4.75 20.78 12.47 0.25 31.17 650.71 

T 8.44 4.87 21.84 13.10 0.25 32.76 667.12 

IU 8.19 4.73 20.60 12.36 0.25 30.90 647.89 

F 8.09 4.67 20.10 12.06 0.25 30.15 639.96 

S 8.23 4.75 20.76 12.46 0.25 31.14 650.37 

[3
1
, 

3
2
] 

H 28.70 16.71 249.04 154.27 0.24 383.59 2287.08 

T 35.81 20.86 387.68 240.19 0.24 597.23 2853.78 

IU 36.94 21.52 412.74 255.72 0.24 635.84 2944.58 

F 36.94 21.52 412.69 255.68 0.24 635.75 2944.39 

S 33.32 19.41 335.73 208.00 0.24 517.19 2655.68 

[2
8
] 

H 8.14 2.53 31.84 3.53 0.45 10.22 355.28 

T 8.16 2.57 31.92 3.66 0.44 10.57 361.53 

IU 8.16 2.61 31.80 3.76 0.44 10.85 366.48 

F 8.17 2.62 31.83 3.80 0.44 10.97 368.47 

S 8.26 2.70 32.25 4.04 0.44 11.63 379.60 

O
th

er
s-

[3
0
] 

V
o
ra

 [
5
] 

H 3.86 2.23 5.98 3.59 0.25 8.98 333.79 

T 3.39 1.96 4.61 2.77 0.25 6.92 293.04 

IU 3.67 2.12 5.43 3.26 0.25 8.15 318.01 

F 3.66 2.11 5.39 3.23 0.25 8.09 316.78 

S 1.66 0.96 1.11 0.67 0.25 1.67 143.82 

O
th

er
s-

T
[3

1
,3

2
] 

V
o
ra

 [
5
] 

H 4.85 2.08 12.87 3.14 0.39 8.72 317.80 

T 4.69 2.26 11.04 3.69 0.35 9.96 342.58 

IU 4.91 2.27 12.47 3.75 0.36 10.22 345.93 

F 4.94 2.31 12.55 3.86 3.36 10.50 350.98 

S 3.30 1.89 4.42 2.59 0.25 6.50 283.61 

O
th

er
s-

[2
8
] 

 

V
o

ra
 [

5
] 

H 9.49 3.56 53.06 9.16 0.42 25.99 545.01 

T 9.61 3.71 53.64 9.95 0.41 28.11 567.51 

IU 9.58 3.67 53.41 9.74 0.41 27.53 561.47 

F 9.57 3.67 53.38 9.73 0.41 27.53 561.41 

H 9.65 3.74 53.86 10.13 0.41 28.60 572.69 

Others 

[21] 

--- --- 13.80 

17.40 

7.60 

5.00 

--- --- 169.00 

169.30 

Expt. [21] --- --- 14.10 6.50 --- --- --- 



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

45 
 

 

Table 6 : Thermodynamic and elastic properties of Zr67Ni33 BMG. 

  

A
p

p
ro

a
ch

 

S
cr

. 
F

u
n

. 𝒗𝑳 × 𝟏𝟎
𝟓 

(𝐜𝐦/𝐬) 

𝒗𝑻 × 𝟏𝟎
𝟓 

(𝐜𝐦/𝐬) 

𝑩𝑻 × 𝟏𝟎
𝟏𝟏 

(𝐝𝐲𝐧𝐞/𝒔𝟐) 

𝑮 × 𝟏𝟎𝟏𝟏 

(𝐝𝐲𝐧𝐞/𝒔𝟐) 

σ 𝒀 × 𝟏𝟎𝟏𝟏 

(𝐝𝐲𝐧𝐞/𝒔𝟐) 

𝜽𝑫   K 

[3
0

] 

H 5.49 3.17 11.67 7.00 0.25 17.51 391.75 

T 5.50 3.18 11.70 7.02 0.25 17.55 392.25 

IU 5.47 3.16 11.59 6.95 0.25 17.38 390.34 

F 5.55 3.20 11.91 7.15 0.25 17.87 395.71 

S 5.49 3.17 11.66 7.00 0.25 17.49 391.51 

[3
1

,3
2

] 

H 19.61 11.51 144.94 92.21 0.24 228.22 1419.31 

T 25.51 14.99 244.58 156.50 0.24 386.96 1848.80 

IU 26.60 15.63 265.81 170.20 0.24 420.78 1927.97 

F 26.61 15.64 265.92 170.26 0.24 420.95 1928.35 

S 23.36 13.72 205.16 131.07 0.24 324.18 1692.01 

[2
8

] 

H 7.61 3.65 27.93 9.29 0.35 25.09 456.96 

T 7.65 3.69 28.09 9.50 0.35 25.61 461.92 

IU 7.68 3.74 28.09 9.74 0.34 26.20 467.54 

F 7.70 3.75 28.17 9.82 0.34 26.38 469.27 

S 7.72 3.76 28.40 9.85 0.34 26.48 470.03 

Others [14] 4.46 1.88 10.40 2.43 --- --- 236.00 

Others-MD 

[15,16] 

4.10 

4.75 

1.95 

2.02 

7.80 

11.72 

2.80 --- --- 217.00 

254±11 

Others-[14,30] 5.00 2.50 11.76 --- --- --- 311.00 

Others- 

[14, 28] 

4.19 2.05 8.46 --- --- --- 256.00 

Others [19] 3.53 2.04 4.94 2.96 --- 7.40 --- 

O
th

er
s-

[3
0

] 

V
o

ra
 [

2
,3

] 

H 1.79 1.03 1.25 0.75 0.25 1.87 127.87 

T 1.81 1.04 1.27 0.76 0.25 1.91 129.23 

IU 0.87 0.50 0.29 0.18 0.25 0.44 62.10 

F 0.85 0.49 0.28 0.17 0.25 0.42 60.80 

S 1.59 0.92 0.99 0.59 0.25 1.48 113.73 

O
th

er
s-

[3
1

,3
2

] 

V
o

ra
 [

2
,3

] 

H 2.41 1.37 2.34 1.31 0.26 3.30 169.30 

T 2.80 1.61 3.06 1.83 0.25 4.57 199.80 

IU 2.72 1.70 2.49 2.03 0.17 5.02 214.60 

F 2.78 1.75 2.55 2.14 0.17 5.02 214.60 

S 2.40 1.48 2.01 1.53 0.20 3.66 181.70 

O
th

er
s-

[2
8

] 
 

V
o

ra
 [

2
,3

] 

H 7.18 3.44 25.02 8.32 0.35 22.46 431.84 

T 7.30 3.57 25.45 8.93 0.34 23.97 446.91 

IU 7.25 3.52 25.28 8.69 0.35 23.38 441.08 

F 7.23 3.50 25.14 8.60 0.35 23.16 438.94 

S 7.36 3.62 25.75 9.21 0.34 24.68 453.72 

 

 

 
       

  



Aditya M. Vora, Alkesh L. Gandhi / BIBECHANA 18 (1) (2021) 33-47  

46 
 

4. Conclusion 
 

Lastly, the computed PDCs of the BMGs have 

satisfactorily reproduced the general characteristic 

features of a broad range of collective excitations. 

The presently computed PDCs and their concerned 

properties through HB, TG and BS approaches 

have shown consistent results with the successful 

application of the Shaw’s model pseudopotential, 

where screening effects are also observed by 

various local field correction functions, which 

supports the PAA model and pseudopotential 

concept. The screening effect plays a significant 

role in the estimation of the thermodynamic and 

elastic properties of said BMGs. The relative 

exchange and correlation effects in the selected 

properties are successfully examined by H-, T-, IU-

, F- and S- local filed correction functions, which 

show variations according to the vibrational 

properties. The calculated properties observed at 

room temperature are found in qualitative 

agreement with acceptable differences with the data 

available in the literature. The PDC generated from 

the three approaches are satisfactorily reproduced 

general characteristics of dispersion curves. The 

thermodynamic and elastic properties obtained due 

to TG [31, 32]-approach are higher than those due 

to HB [30] or BS [31, 32]-approaches. Hence, the 

pseudopotential theory is remarkably helpful for 

the framework of computation of theoretical data 

for a specific BMG. 

 

References 
 

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