Bulk Metallic Glasses [1 ed.] 9781621001638, 9781611229387

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PHYSICS RESEARCH AND TECHNOLOGY

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BULK METALLIC GLASSES

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PHYSICS RESEARCH AND TECHNOLOGY

BULK METALLIC GLASSES

THOMAS F. GEORGE RENAT R. LETFULLIN AND

GUOPING ZHANG Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Bulk metallic glasses / editors, Thomas F. George, Renat R. Letfullin, and Guoping Zhang. p. cm. -- (Physics research and technology) Includes bibliographical references and index. ISBN HERRN 1. Metallic glasses--Electric properties. 2. Bulk solids. 3. Superconductivity. 4. Electron-phonon interactions. 5. Electromagnetic compatibility. I. George, Thomas F., 1947- II. Letfullin, Renat R. III. Zhang, Guoping, 1970QC611.98.A44B85 2011 530.4'13--dc22 2010046844

Published by Nova Science Publishers, Inc. † New York

CONTENTS vii 

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Preface Chapter 1

Superconducting State Parameters of Bulk Metallic Glasses Aditya M. Vora 

Chapter 2

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass Aditya M. Vora 

15 

Chapter 3

Study of Vibrational Dynamics of Binary Mg70Zn30 Metallic Glass Aditya M. Vora 

35 

Chapter 4

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass Aditya M. Vora 

45 

Chapter 5

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass Aditya M. Vora 

65 

Chapter 6

Computation of Phonon Dispersion Curves of Ca70Mg30 Metallic Glass Aditya M. Vora 

85 

Chapter 7

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass Aditya M. Vora 

95 

Chapter 8

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass Aditya M. Vora 

115 

Chapter 9

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass 137  Aditya M. Vora 

Index

1 

157 

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PREFACE An amorphous metal is a metallic material with a disordered atomic-scale structure. In contrast to most metals, which are crystalline and therefore have a highly ordered arrangement of atoms, amorphous alloys are non-crystalline. This new book presents and reviews research in the study of bulk metallic glasses. Chapter 1 - The screening dependence theoretical investigations of the superconducting state parameters (SSP) viz. the electron-phonon coupling strength λ , the Coulomb pseudopotential

μ * , the transition temperature TC , the isotope effect exponent α and the

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effective interaction strength N OV of some bulk metallic glasses (BMG) viz. Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 have been reported for the first time using Ashcroft’s empty core (EMC) model potential. Five local field correction functions proposed by Hartree (H), Taylor (T), Ichimaru-Utsumi (IU), Farid et al. (F) and Sarkar et al. (S) are used in the present investigation to study the screening influence on the aforesaid properties. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential

μ * , the isotope effect exponent α and the effective interaction strength

N OV show weak dependences on the local field correction functions. The transition temperature TC obtained from H-local field correction function is found an excellent agreement with available experimental data. Also, the present results are found in qualitative agreement with other such earlier reported data, which confirms the superconducting phase in the bulk metallic glasses (BMG). Chapter 2 - The vibrational dynamics of Pd77.5Si16.5Cu6 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic

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viii

Thomas F. George, Renat R. Letfullin and Guoping Zhang

properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloy-atom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 3 - The vibrational dynamics of Mg70Zn30 metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 4 - The vibrational dynamics of Fe80B14Si6 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 5 - The vibrational dynamics of Fe80B10Si10 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), TakenoGoda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloy-atom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 6 - The computation of the phonon dispersion curves of Ca70Mg30 metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). Our recently proposed model potential is employed successfully to explain electron-ion interaction in the metallic glass. The local field correction function due to Sarkar et al is used for the first time to introduce the exchange and correlation effects in the aforesaid properties. The effective pair potential is used to generate the pair correlation function g(r). The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic and elastic properties viz. longitudinal and transverse sound velocities, isothermal bulk modulus, modulus of rigidity, Poisson’s ratio, Young’s modulus and Debye temperature are also investigated successfully.

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Preface

ix

Chapter 7 - The vibrational dynamics of Fe40Ni40B20 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), TakenoGoda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloy-atom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 8 - The vibrational dynamics of Ni80B10Si20 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), TakenoGoda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloy-atom (PAA) model is applied for the first time instead of Vegard's Law. Chapter 9 - The vibrational dynamics of Fe60Ni20B10Si10 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloy-atom (PAA) model is applied for the first time instead of Vegard's Law. Versions of these chapters were also published in International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics, Volume 3, Numbers 1-4, edited by Renat R. Letfullin, Guoping Zhang, and Thomas F. George, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

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In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 1-14 © 2011 Nova Science Publishers, Inc.

Chapter 1

SUPERCONDUCTING STATE PARAMETERS OF BULK METALLIC GLASSES Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj – Kutch, 370 001, Gujarat, India

ABSTRACT The screening dependence theoretical investigations of the superconducting state parameters (SSP) viz. the electron-phonon coupling strength λ , the Coulomb pseudopotential

μ* ,

the transition temperature

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interaction strength

N OV

TC ,

the isotope effect exponent

α

and the effective

of some bulk metallic glasses (BMG) viz. Ti50Be34Zr10,

(Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 have been reported for the first time using Ashcroft’s empty core (EMC) model potential. Five local field correction functions proposed by Hartree (H), Taylor (T), Ichimaru-Utsumi (IU), Farid et al. (F) and Sarkar et al. (S) are used in the present investigation to study the screening influence on the aforesaid properties. It is observed that the electron-phonon coupling strength λ and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential

N OV

μ* ,

the isotope effect exponent

α

and the effective interaction strength

show weak dependences on the local field correction functions. The transition

temperature

TC

obtained from H-local field correction function is found an excellent

agreement with available experimental data. Also, the present results are found in qualitative agreement with other such earlier reported data, which confirms the superconducting phase in the bulk metallic glasses (BMG).

Keywords: Pseudopotential; superconducting state parameters; bulk metallic glasses (BMG). PACS Number(s): 61.43.Dq; 71.15.Dx; 74.20.-z; 74.70.Ad ∗

E-mail address: [email protected]. Tel.: +91-2832-256424.

2

Aditya M. Vora

1. INTRODUCTION During last several years, the superconductivity remains a dynamic area of research in condensed matter physics with continual discoveries of novel materials and with an increasing demand for novel devices for sophisticated technological applications. A large number of metals and amorphous alloys are superconductors, with critical temperature TC ranging from 1-18K [1-11]. The pseudopotential theory has been used successfully in explaining the superconducting state parameters (SSP) for metallic complexes by many workers [3-11]. Many of them have used well known model pseudopotential in the calculation of the SSP for the metallic complexes. Recently, we have studied the superconducting state parameters (SSP) of some metallic superconductors based on the various elements of the periodic table using single parametric model potential formalism [311]. The study of the superconducting state parameters (SSP) of the bulk metallic glasses (BMG)may be of great help in deciding their applications; the study of the dependence of the transition temperature TC on the composition of metallic elements is helpful in finding new

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superconductors with high transition temperature TC . The application of pseudopotential to a ternary system involves the assumption of pseudoions with average properties, which are assumed to replace three types of ions in the ternary systems, and a gas of free electrons is assumed to permeate through them. The electron-pseudoion is accounted for by the pseudopotential and the electron-electron interaction is involved through a dielectric screening function. For successful prediction of the superconducting properties of the alloying systems, the proper selection of the pseudopotential and screening function is very essential [3-11]. Therefore, in the present article, we have used the well known McMillan’s theory [12] of the superconductivity for predicting the superconducting state parameters (SSP) of some Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 bulk metallic glasses (BMG) for the first time. We have used Ashcroft’s empty core (EMC) model potential [13] for studying the electron-phonon coupling strength

λ , Coulomb pseudopotential μ * , transition

temperature TC , isotope effect exponent α and effective interaction strength N OV for the first time. To see the impact of various exchange and correlation functions on the aforesaid properties, we have employed here five different types of local field correction functions proposed by Hartree (H) [14], Taylor (T) [15], Ichimaru-Utsumi (IU) [16], Farid et al. (F) [17] and Sarkar et al. (S) [18]. We have incorporated for the first time more advanced local field correction functions due to IU [16], F [17] and S [18] with EMC model potential [13] in the present computation of the SSP for bulk metallic glasses (BMG). In the present work, the pseudo-alloy-atom (PAA) model was used to explain electronion interaction for alloying systems [3-11]. It is well known that the pseudo-alloy-atom (PAA) model is a more meaningful approach to explain such kind of interactions in alloying systems. In the PAA approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, which occupy the lattice sites and form a perfect lattice in the same way as pure metals. In this model the hypothetical crystal made up of PAA is supposed to

Superconducting State Parameters of Bulk Metallic Glasses

3

have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy and metallic glass. The complete miscibility in the alloy systems is considered as a rare case. Therefore, in such alloying systems the atomic matrix elements in the pure states are affected by the characteristics of alloys such as lattice distortion effects and charging effects. In the PAA model, such effects are involved implicitly. In addition to this it also takes into account the self-consistent treatment implicitly. Looking to the advantage of the PAA model, we propose a use of PAA model for the first time to investigate the SSP of bulk metallic glasses (BMG).

2. THEORY In the present investigation for bulk metallic glasses (BMG), the electron-phonon coupling strength λ is computed using the relation [3-11]

λ

=

mb Ω0 2 4π k F M 〈 ω 2 〉

2k F

∫q

3

W (q ) dq . 2

(1)

0

Here mb is the band mass, M the ionic mass, ΩO the atomic volume, k F the Fermi wave vector, W (q ) the form factor and

ω 2 the averaged square phonon frequency, of the

ternary metallic glasses, respectively. The Butler [19],

ω2

12

ω 2 is calculated using the relation given by

= 0.69 θ D , where θ D is the Debye temperature of the bulk metallic

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glasses (BMG). Using X = q 2k F and Ω O = 3 π Z 2

λ

=

12 m b Z M 〈ω 〉 2

(k F )3 , we get Eq. (1) in the following form, 1

∫X

3

W (X )

2

dX ,

(2)

0

where Z and W (X ) are the valence and the EMC form factor [13] of the bulk metallic glasses (BMG), respectively. The well known Ashcroft’s empty core (EMC) model potential [13] form factor used in the present computations of the SSP of bulk metallic glasses (BMG) is of the form,

W (X ) =

− 2πZ cos(2k F XrC ) , ΩO X 2 k F2 ε ( X )

(3)

here rC is the parameter of the model potential of bulk metallic glasses (BMG). The Ashcroft’s empty core (EMC) model potential is a simple one-parameter model potential

4

Aditya M. Vora

[13], which has been successfully found for various metallic complexes [5-8]. When used with a suitable form of dielectric screening functions, this potential has also been found to yield good results in computing the SSP of metallic complexes [5-8]. Therefore, in the present work we use Ashcroft’s empty core (EMC) model potential [13] with more advanced Ichimaru-Utsumi (IU) [16], Farid et al. (F) [17] and Sarkar et al. (S) [18] local field correction functions for the first time. The model potential parameter rC may be obtained by fitting either to some experimental data or to realistic form factors or other data relevant to the properties to be investigated. In the present work, rC is fitted with experimental TC [20] of the bulk metallic glasses (BMG) for most of the local field correction functions. The Coulomb pseudopotential

μ* =

μ * is given by [3-11] mb π kF

1

dX

∫ ε (X ) 0

⎛ EF mb 1+ ln ⎜⎜ π k F ⎝ 10 θ D

⎞ ⎟⎟ ⎠

1

∫ 0

dX ε (X )

.

(4)

Where EF is the Fermi energy, m b the band mass of the electron, θ D the Debye

temperature and ε (X ) the modified Hartree dielectric function, which is written as [14]

ε(X ) = 1 + (ε H (X ) − 1) (1 − f (X )) .

(5)

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Here ε H (X ) is the static Hartree dielectric function and the expression of it is given by [14], ε H (X ) = 1 +

⎞ ⎛ 1 − η2 m e2 1+ η ⎟ ;η = q . ⎜ ln + 1 ⎟ ⎜ 2η 2 2 1 − η 2k F 2π k = η F

⎝

(6)

⎠

While f (X ) is the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [14], Taylor (T) [15], Ichimaru-Utsumi (IU) [16], Farid et al. (F) [17] and Sarkar et al. (S) [18] are incorporated to see the impact of exchange and correlation effects. The details of all the local field corrections are below. The Hartree (H) screening function [14] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(7)

Taylor (T) [15] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [15],

Superconducting State Parameters of Bulk Metallic Glasses

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ . ⎢1 + 2 ⎥ π k F ⎦ ⎣

5

(8)

The Ichimaru-Utsumi (IU) local field correction function [16] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is

⎡ ⎛q⎞ ⎛ ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ 3 ⎠ ⎝ kF ⎠ ⎢⎣ ⎝ kF ⎠ ⎝ ⎝ kF ⎠ ⎝ kF ⎠ 4

2

4

2

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪4 − ⎜ ⎟ 2 + ⎜⎜ ⎟⎟ ⎪ ⎤⎪ ⎜ k ⎟ ⎝ kF ⎠ ⎪ . − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎥⎦ ⎪ ⎛ q ⎞ 4⎜ ⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩ (9)

On the basis of Ichimaru-Utsumi (IU) local field correction function [16] local field correction function, Farid et al. (F) [17] have given a local field correction function of the form

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⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎤⎪ ⎡ ⎛q⎞ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎢⎣ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 4 2 − ⎜⎜ ⎟⎟ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(10)

Based on Eqs. (9-10), Sarkar et al. (S) [18] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(11)

The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [16-18]. After evaluating

λ and μ * , the transition temperature TC and isotope effect exponent

α are investigated from the McMillan’s formula [3-11, 12] TC =

⎡ − 1.04(1 + λ ) ⎤ θD exp⎢ ⎥, * 1.45 ⎣ λ − μ (1 + 0.62λ )⎦

(12)

6

Aditya M. Vora

α=

θD 1⎡ ⎛ * ⎢1 − ⎜ μ ln 2 ⎢ ⎜⎝ 1.45TC ⎣

2 ⎞ 1 + 0.62λ ⎤ ⎟⎟ ⎥. ⎠ 1.04(1 + λ )⎥⎦

(13)

The expression for the effective interaction strength N OV is studied using [4-9]

N OV =

λ − μ* . 10 1+ λ 11

(14)

3. RESULTS AND DISCUSSION The values of the input parameters for the some bulk metallic glasses (BMG) viz. Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 under investigation are assembled in Table 1. To determine the input parameters and various constants for PAA model [3-11], the following definitions for bulk metallic glasses (BMG) Ax B y C z ( x + y + z = 1) are

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adopted,

Z = x(Z A ) + y (Z B ) + z (Zc ) ,

(15)

M = x(M A ) + y (M B ) + z (Mc ) ,

(16)

Ω O = x(Ω O A ) + y (Ω O B ) + z (Ω O C ) ,

(17)

rC = x(rCA ) + y (rCB ) + z (rCC ) ,

(18)

θ D = x(θ DA ) + y (θ DB ) + z (θ DC ) .

(20)

Where x , y and z are the concentration factors of the A , B and C pure metallic components of bulk metallic glasses (BMG). The input parameters of the pure metallic components are taken from the literature [14, 21]. The presently calculated results of the superconducting state parameters (SSP) of bulk metallic glasses (BMG) are tabulated in Table 2 with the other such experimental findings [20].

Superconducting State Parameters of Bulk Metallic Glasses

7

Table 1. Input parameters and other constants of bulk metallic glasses (BMG) Ternary Metallic Glasses Ti50Be34Zr10 (Mo0.6Ru0.4)78B22 (Mo0.6Ru0.4)80B20 (Mo0.4Ru0.6)80P20 (Mo0.6Ru0.4)70Si30 (Mo0.6Ru0.4)84B16 (Mo0.6Ru0.4)72Si28 (Mo0.6Ru0.4)86B14 (Mo0.6Ru0.4)76Si24 (Mo0.6Ru0.4)78Si22 (Mo0.6Ru0.4)80Si20 (Mo0.6Ru0.4)82Si18 (Mo0.6Ru0.4)80P20

Z

rC (au)

Ω O (au)3

θ D (K)

2.70 4.72 4.76 4.84 4.84 4.85 4.86 4.89 4.91 4.94 4.96 4.98 5.16

0.8798 0.5894 0.5843 0.5908 0.6692 0.5205 0.6616 0.5314 0.6544 0.6444 0.6407 0.6353 0.5746

96.42 88.65 89.66 107.10 110.13 91.68 109.44 92.69 108.06 107.37 106.68 105.99 109.26

190.00 280.00 277.00 267.00 554.70 301.00 552.12 295.00 546.96 544.38 541.80 539.22 265.00

Table 2. Superconducting state parameters of bulk metallic glasses (BMG) Present results

Ternary Metallic Glasses

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Ti50Be34Zr10

(Mo0.6Ru0.4)78B22

(Mo0.6Ru0.4)80B20

SSP

Expt. [20] H

T

IU

F

S

λ

0.3813

0.5099

0.5319

0.5328

0.4615

0.40

μ*

0.1201

0.1290

0.1302

0.1304

0.1251

−

TC (K)

0.2743

1.2943

1.5441

1.5506

0.8338

0.274

α

0.2639

0.3512

0.3604

0.3602

0.3305

−

NOV

0.1940

0.2602

0.2708

0.2711

0.2370

−

λ

0.6321

0.8295

0.8623

0.8655

0.7245

−

μ*

0.1131

0.1206

0.1216

0.1218

0.1165

−

TC (K)

5.4038 10.1946 10.9880 11.0608

7.6583 5.40, 5.42

α

0.4329

0.4499

0.4519

0.4520

0.4429

−

NOV

0.3296

0.4041

0.4152

0.4162

0.3666

−

λ

0.6475

0.8507

0.8844

0.8879

0.7422

−

μ*

0.1130

0.1205

0.1215

0.1217

0.1164

−

TC (K)

5.7510 10.6463 11.4495 11.5255

8.0532

5.75

α

0.4359

0.4520

0.4538

0.4539

0.4453

−

NOV

0.3364

0.4118

0.4229

0.4240

0.3737

−

8

Aditya M. Vora Table 2. (Continued) Ternary Metallic Glasses

(Mo0.6Ru0.4)84B16

(Mo0.6Ru0.4)86B14

(Mo0.4Ru0.6)80P20

SSP

T

IU

F

S

λ μ*

0.6554

0.8690

0.9044

0.9091

0.7488

−

0.1142

0.1218

0.1228

0.1230

0.1176

−

TC (K) α

6.4006 11.9833 12.8916 13.0077

8.8574

6.40

0.4356

0.4522

0.4541

0.4543

0.4446

−

NOV λ μ*

0.3392

0.4174

0.4289

0.4304

0.3756

−

0.6542

0.8665

0.9017

0.9063

0.7481

−

0.1140

0.1216

0.1226

0.1228

0.1174

−

TC (K) α

6.2534 11.6941 12.5811 12.6900

8.6755

6.25

0.4357

0.4522

0.4541

0.4543

0.4448

−

NOV λ μ*

0.3388

0.4167

0.4282

0.4296

0.3754

−

0.6183

0.8252

0.8603

0.8642

0.7187

0.65

0.1156

0.1235

0.1246

0.1248

0.1195

−

TC (K)

4.6833

9.4181

10.2240 10.3100

6.9723

4.68

0.4260

0.4463

0.4486

0.4488

0.4381

−

0.3218

0.4009

0.4128

0.4141

0.3624

−

0.7226

0.9622

1.0026

1.0073

0.8361 0.65, 0.71

0.1147

0.1224

0.1235

0.1237

0.1184

−

7.3102 12.7635 13.6323 13.7281

9.9639

6.0, 7.31

0.4450

0.4585

0.4600

0.4601

0.4528

−

0.3669

0.4479

0.4599

0.4613

0.4078

−

0.4893

0.6465

0.6731

0.6752

0.5716

−

0.1269

0.1366

0.1379

0.1381

0.1317

−

TC (K) α

3.2042

8.8915

10.0186 10.0944

5.9950

3.20

0.3450

0.3920

0.3973

0.3974

0.3760

−

NOV

0.2508

0.3212

0.3321

0.3328

0.2895

−

λ μ* TC (K) α NOV

0.5008

0.6620

0.6892

0.6914

0.5845

−

0.1266

0.1362

0.1375

0.1377

0.1313

−

3.6039

9.6184

10.7901 10.8723

6.5512

3.60

0.3539

0.3976

0.4025

0.4026

0.3824

−

0.2572

0.3282

0.3392

0.3400

0.2959

−

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TC (K)

α NOV λ μ*

(Mo0.6Ru0.4)70Si30

(Mo0.6Ru0.4)72Si28

Expt. [20]

H

α NOV λ μ*

(Mo0.6Ru0.4)80P20

Present results

Superconducting State Parameters of Bulk Metallic Glasses

9

Table 2. (Continued) Ternary Metallic Glasses

SSP

(Mo0.6Ru0.4)78Si22

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(Mo0.6Ru0.4)80Si20

(Mo0.6Ru0.4)82Si18

Expt. [20]

H

T

IU

F

S

0.5131

0.6773

0.7050

0.7073

0.5975

−

0.1259

0.1354

0.1367

0.1369

0.1305

−

TC (K) α

4.0523 10.3443 11.5459 11.6318

7.1304

4.05

0.3635

0.4035

0.4080

0.4081

0.3894

−

NOV λ μ*

0.2640

0.3354

0.3463

0.3472

0.3026

−

0.5315

0.7019

0.7306

0.7331

0.6181

−

0.1256

0.1350

0.1362

0.1365

0.1301

−

TC (K) α

4.7547 11.5329 12.7978 12.8935

8.0647

4.75

0.3744

0.4104

0.4145

0.4146

0.3974

−

NOV λ μ*

0.2737

0.3461

0.3572

0.3580

0.3125

−

0.5379

0.7101

0.7390

0.7416

0.6250

−

0.1252

0.1346

0.1359

0.1361

0.1297

−

TC (K)

5.0087 11.9174 13.1954 13.2931

8.3770

5.0

0.3784

0.4129

0.4168

0.4169

0.4003

−

0.2772

0.3497

0.3608

0.3617

0.3158

−

λ μ* (Mo0.6Ru0.4)76Si24

Present results

α NOV λ μ*

0.5477

0.7229

0.7523

0.7549

0.6357

−

0.1249

0.1342

0.1355

0.1357

0.1294

−

TC (K)

5.4019 12.5274 13.8300 13.9319

8.8687

5.40

0.3837

0.4163

0.4200

0.4201

0.4043

−

0.2823

0.3552

0.3663

0.3672

0.3209

−

α NOV

The calculated values of the electron-phonon coupling strength λ for bulk metallic glasses (BMG), using five different types of the local field correction functions with EMC model potential, are shown in Table 2 with the experimental data [20]. It is noticed from the present study that, the percentile influence of the various local field correction functions with respect to the static H-screening function on the electron-phonon coupling strength λ is 18.45%-35.65%, 14.62%-36.92%, 14.25%-38.71%, 14.35%-38.54%, 16.24%-39.77%, 15.71%-39.40%, 16.82%-37.99%, 16.71%-37.99%, 16.71%-38.06%, 16.45%-38.06%, 16.45%-37.85%, 16.29%-37.93% and 16.07%-37.83% for Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 ternary metallic glasses, respectively. Also, the H-screening yields lowest values of the electron-phonon coupling strength λ , whereas the values obtained from the F-function are the highest. It is also observed from the Table 2 that, the electron-phonon

10

Aditya M. Vora

coupling strength λ goes increasing for (Mo0.6Ru0.4)1-xBx and (Mo0.6Ru0.4)1-xSix bulk metallic glasses (BMG) as the concentration ‘ x ’ of the third metallic elements decreases. The increase in the electron-phonon coupling strength λ with concentration ‘ x ’ of the third metallic elements shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour of electrons and phonons, which may be attributed to an increase of the hybridization of sp-d electrons of the third metallic elements with increasing concentration ( x ). This may also be attributed to the increase role of ionic vibrations in the third metallic elements-rich region. The computed values of the Coulomb pseudopotential

μ * , which accounts for the

Coulomb interaction between the conduction electrons, obtained from the various forms of the local field correction functions are tabulated in Table 2. It is observed from the Table 2 that for all the bulk metallic glasses (BMG), the Coulomb pseudopotential

μ * lies between

0.11 and 0.14, which is in accordance with McMillan [12], who suggested that the Coulomb pseudopotential μ ≈ 0.13 for transition metals. The weak screening influence shows on the *

computed values of the Coulomb pseudopotential

μ * . The percentile influence of the various

local field correction functions with respect to the static H-screening function on the Coulomb pseudopotential μ for the bulk metallic glasses (BMG) is observed in the range of 4.16%*

8.58%, 3.01%-7.69%, 3.01%-7.10%, 2.98%-7.71%, 2.98%-7.72%, 3.37%-7.96%, 3.23%7.85%, 3.78%-8.83%, 3.71%-8.77%, 3.65%-8.74%, 3.58%-8.68%, 3.59%-8.71% and 3.60%8.71% for Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 bulk metallic glasses (BMG), respectively. Again the H-screening function yields lowest values of the Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Coulomb pseudopotential

μ * , while the values obtained from the F-function are the highest.

Table 2 contains calculated values of the transition temperature TC for bulk metallic glasses (BMG) computed from the various forms of the local field correction functions along with the experimental findings [20]. From the Table 2 it is noted that, the static H-screening function yields lowest transition temperature TC whereas the F-function yields highest values of transition temperature TC . The present results obtained from the H-local field correction functions are found in good agreement with available experimental data [20]. The theoretical data of transition temperature TC for most of the bulk metallic glasses (BMG) is not available in the literature. The calculated results of the transition temperature TC for Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20 bulk metallic glasses (BMG) deviate in the range of 51.75%-539.56%, 0.07%-114.83%, 0.02%-100.44%, 0.03%-103.25%, 0.05%-103.04%, 0.07%-120.30%, 0.0%-87.80%, 0.13%-215.45%, 0.11%-202.01%, 0.06%187.20%, 0.10%-171.44%, 0.17%-165.86% and 0.04%-158.0% from the experimental findings [20], respectively.

Superconducting State Parameters of Bulk Metallic Glasses

11

It is also noted from the Table 2 that the transition temperature TC increases for (Mo0.6Ru0.4)78B22 and (Mo0.6Ru0.4)1-xSix bulk metallic glasses (BMG) as the concentration ‘ x ’ of the third metallic elements decreases. While the transition temperature TC computed from H-local field correction function for Ti50Be34Zr10 bulk metallic glass is found in good agreement with the experimental data [20]. The presently computed values of the transition temperature TC are found in the range, which is suitable for further exploring the applications of the bulk metallic glasses (BMG) for usage like lossless transmission line for cryogenic applications. While alloying elements show good elasticity and could be drawn in the form of wires as such they have good chances of being used as superconducting transmission lines at low temperature of the order of 7K. The values of the isotope effect exponent α for bulk metallic glasses (BMG) are tabulated in Table 2. The computed values of the isotope effect exponent α show a weak dependence on the dielectric screening, its value is being lowest for the H-screening function and highest for the F-function. Since the experimental value of the isotope effect exponent α has not been reported in the literature so far, the present data of the isotope effect exponent α may be used for the study of ionic vibrations in the superconductivity of alloying substances. Since H-local field correction function yields the best results for the electronphonon coupling strength λ and the transition temperature TC , it may be observed that the isotope effect exponent α values obtained from this screening provide the best account for the role of the ionic vibrations in superconducting behaviour of this system. The values of the effective interaction strength N OV are listed in Table 2 for different

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

local field correction functions. It is observed that the magnitude of the effective interaction strength N OV shows that the bulk metallic glasses (BMG) under investigation lie in the range of weak coupling superconductors. The values of the effective interaction strength N OV also show a feeble dependence on dielectric screening, its value being lowest for the Hscreening function and highest for the F-screening function. The variation of present values of the effective interaction strength N OV show that, the bulk metallic glasses (BMG) under consideration fall in the range of weak coupling superconductors. The effect of local field correction functions plays an important role in the computation

μ * , which makes drastic variation on the transition temperature TC , the isotope effect exponent α and of the electron-phonon coupling strength λ and the Coulomb pseudopotential

the effective interaction strength N OV . The local field correction functions due to IU, F and S are able to generate consistent results regarding the SSP of Ti50Be34Zr10, (Mo0.6Ru0.4)78B22, (Mo0.6Ru0.4)80B20, (Mo0.4Ru0.6)80P20, (Mo0.6Ru0.4)70Si30, (Mo0.6Ru0.4)84B16, (Mo0.6Ru0.4)72Si28, (Mo0.6Ru0.4)86B14, (Mo0.6Ru0.4)76Si24, (Mo0.6Ru0.4)78Si22, (Mo0.6Ru0.4)80Si20, (Mo0.6Ru0.4)82Si18 and (Mo0.6Ru0.4)80P20bulk metallic glasses (BMG) as those obtained from more commonly employed H and T local field correction functions. Thus, the use of these more promising local field correction functions is established successfully. The computed results of the isotope effect exponent α and the effective interaction strength N OV are not showing any abnormal values for the bulk metallic glasses (BMG).

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

12

Aditya M. Vora

Also, the main differences of the local field correction functions are played in important role in the production of the SSP of bulk metallic glasses (BMG). The Hartree (H) dielectric function [14] is purely static and it does not include the exchange and correlation effects. Taylor (T) [15] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. The Ichimaru-Utsumi (IU) local field correction function [16] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also satisfies the self consistency condition in the compressibility sum rule and short range correlations. Therefore, Hartree (H) local field correction function [14] gives the best agreement with the experiment with EMC model potential and found suitable in the present case. On the basis of Ichimaru-Utsumi (IU) local field correction function [16], Farid et al. (F) [17] and Sarkar et al. [18] have given a local field correction function. Hence, IU- and F-functions represent same characteristic nature of the SSP. Also, the SSP computed from Sarkar et al. [18] local field correction are found in qualitative agreement with the available experimental data [20]. The numerical values of the aforesaid properties are found to be quite sensitive to the selection of the local field correction function and showing a significant variation with the change in the function. Thus, the use of these more promising local field correction functions is established successfully. The theoretical data for SSP for ternary superconductors are not available in the literature for detailed comparison, but the comparison with other such theoretical values is encouraging, which confirms the applicability of EMC model potential in explaining the superconducting state parameters of bulk metallic glasses (BMG). According to Matthias rules [22, 23] the alloys having Z>2 exhibit superconducting nature. Hence, the presently computed bulk metallic glasses (BMG) are the superconductors. Also, for (Mo0.6Ru0.4)100-xBx bulk metallic glasses (BMG), when we go from Z=4.72 to Z=4.89 and for (Mo0.6Ru0.4)100-xSix bulk metallic glasses (BMG) when we go from Z=4.84 to Z=4.98, the electron-phonon coupling strength λ changes with lattice spacing “a”. Similar trends are also observed in the values of the transition temperature TC for most of the bulk metallic glasses (BMG). Hence, a strong dependency of the superconducting state parameters (SSP) of the bulk metallic glasses (BMG) on the valence Z is found. Also from the presently computed results of the superconducting state parameters (SSP) of bulk metallic glasses (BMG), we are observed that, for (Mo0.6Ru0.4)100-xBx, as the atomic volume ΩO , the superconducting state parameters (SSP) increases while those for and (Mo0.6Ru0.4)100-xSix bulk metallic glasses (BMG) as the atomic volume ΩO decreases, the superconducting state parameters (SSP) increases. Lastly, we would like to emphasize the importance of involving a precise form for the pseudopotential. It must be confessed that although the effect of pseudopotential in strong coupling superconductor is large, yet it plays a decisive role in weak coupling superconductors i.e. those substances which are at the boundary dividing the superconducting and nonsuperconducting region. In other words, a small variation in the value of electron-ion interaction may lead to an abrupt change in the superconducting properties of the material under consideration. In this connection we may realize the importance of an accurate form for the pseudopotential.

Superconducting State Parameters of Bulk Metallic Glasses

13

CONCLUSIONS Lastly we concluded that, the H-local field corrections when used with EMC model potential provide the best explanation for superconductivity in the bulk metallic glasses (BMG). The values of the electron-phonon coupling strength λ and the transition temperature TC show an appreciable dependence on the local field correction function, whereas for the Coulomb pseudopotential

μ * , the isotope effect exponent α and the effective interaction strength

N OV a weak dependence is observed. The magnitude of the electron-phonon coupling strength

λ , the isotope effect exponent α and the effective interaction strength NOV values

shows that, the bulk metallic glasses (BMG) are weak to intermediate superconductors. In the absence of theoretical or experimental data for the isotope effect exponent α and the effective interaction strength N OV , the presently computed values of these parameters may be considered to form reliable data for these ternary systems, as they lie within the theoretical limits of the Eliashberg-McMillan formulation. The comparisons of presently computed results of the superconducting state parameters (SSP) of the bulk metallic glasses (BMG) with available experimental findings are highly encouraging, which confirms the applicability of the EMC model potential and different forms of the local field correction functions. Such study on superconducting state parameters (SSP) of other bulk metallic glasses (BMG) is in progress.

REFERENCES

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[1]

A. V. Narlikar and S. N. Ekbote, Superconductivity and Superconducting Materials (South Asian Publishers New Delhi – Madras, 1983). [2] P. B. Allen, Handbook of Superconductivity, Ed. C. P. Poole, Jr. (Academic Press, New York, 1999) p. 478. [3] A. M. Vora, M. H. Patel, S. R. Mishra, P. N. Gajjar and A. R. Jani, Solid State Phys., 44 (2001) 345. [4] P. N. Gajjar, A. M. Vora and A. R. Jani, Mod. Phys. Lett. B18 (2004) 573. [5] Aditya M. Vora, Physica C 450 (2006) 135; Physica C458 (2007) 21; Physica C458 (2007) 43. [6] Aditya M. Vora, J. Supercond. Novel Magn. 20 (2007) 355; J. Supercond. Novel Magn. 20 (2007) 373; J. Supercond. Novel Magn. 20 (2007) 387; Int. J. Theor. Phys., Group Theo. Non. Opt. 12 (2008) 283; Int. J. Theor. Phys., Group Theo. Non. Opt. 13 (2008) 1. [7] Aditya M. Vora, Phys. Scr. 76 (2007) 204; Frontiers Phys. 2 (2007) 430; J. Optoelec. Adv. Mater. 9 (2007) 2498. [8] Aditya M. Vora, Comp. Mater. Sci. 40 (2007) 492; Chinese Phys. Lett. 24 (2007) 2624. [9] A. M. Vora, M. H. Patel, P. N. Gajjar and A. R. Jani, Pramana-J. Phys. 58 (2002) 849. [10] P. N. Gajjar, A. M. Vora, M. H. Patel and A. R. Jani, Int. J. Mod. Phys. B17 (2003) 6001. [11] P. N. Gajjar, A. M. Vora and A. R. Jani, Indian J. Phys. 78 (2004) 775.

14

W. L. McMillan, Phys. Rev. 167 (1968) 331. N. W. Ashcroft, Phys. Lett. 23 (1966) 48. W. A. Harrison, Elementary Electronic Structure, (World Scientific, Singapore, 1999). R. Taylor, J. Phys. F: Met. Phys. 8 (1978) 1699. S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1981) 7386. B. Farid, V. Heine, G. Engel and I. J. Robertson, Phys. Rev. B48 (1993) 11602. A. Sarkar, D. Sen, H. Haldar and D. Roy, Mod. Phys. Lett. B12 (1998) 639. W. H. Butler, Phys. Rev. B15 (1977) 5267. U. Mizutani, Prog. Mater. Sci. 28 (1983) 97. C. Kittel, Introduction to Solid State Physics, (John Wiley & Sons, Inc., Singapore, 1996) 7th Ed., p. 336. [22] B. T. Matthias: Progress in Low Temperature Physics, Ed. C. J. Gorter (North Holland, Amsterdam, 1957), Vol. 2. [23] B. T. Matthias, Physica 69, 54 (1973).

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[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Aditya M. Vora

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 15-34 © 2011 Nova Science Publishers, Inc.

Chapter 2

STUDY OF VIBRATIONAL DYNAMICS OF PD77.5SI16.5CU6 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, 370 001, Gujarat, India

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ABSTRACT The vibrational dynamics of Pd77.5Si16.5Cu6 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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∗

E-mail address: [email protected]. Tel. : +91-2832-256424

16

Aditya M. Vora

18]. The Pd77.5Si16.5Cu6 is the most important candidate of bulk metallic glasses. The PDC of Pd77.5Si16.5Cu6 glass has been theoretically investigated by Agarwal et al. [11, 12] and by Aziz-Ray [13] using BS approach [17] assuming the force among nearest neighbours as central and volume dependent. The experimental data of PDC of this glass is not available in the literature. Therefore, we have reported vibrational properties of this glass using pseudopotential theory for the first time. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [14], Takeno-Goda (TG) [15, 16] and Bhatia-Singh (BS) [17, 18] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [19], Taylor (T) [20], Ichimaru-Utsumi (IU) [21], Farid et al. (F) [22] and Sarkar et al. (S) [23] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio

σ

and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 24],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ Vs (r ) = ⎜ S F (q ) ⎢ ∫ ⎥ q dq . 2 ⎜ r ⎟ π ⎣ qr ⎦ ⎝ ⎠

(2)

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

17

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [24], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2 16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [19]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [19],

ε H (q ) = 1 +

m e2 2 π k = 2 η2 F

⎛ 1 − η2 ⎞ 1+ η ⎜ ⎟ ;η = q ln + 1 ⎜ 2η ⎟ 1− η 2k F ⎝ ⎠

(5)

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [19], Taylor (T) [20], Ichimaru-Utsumi (IU) [21], Farid et al. (F) [22] and Sarkar et al. (S) [23] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [19] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [20] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [20],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ ⎢1 + ⎥. π k F2 ⎦ ⎣

(7)

18

Aditya M. Vora

The Ichimaru-Utsumi (IU) local field correction function [21] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎛ ⎤⎪ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ . f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 k k k k ⎛ ⎞ ⎛ q q⎞ ⎠⎝ F ⎠ ⎢⎣ ⎝ F ⎠ ⎝ ⎥⎦ ⎪ ⎝ F⎠ ⎝ F⎠ 4⎜⎜ ⎟⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩ (8) On the basis of Ichimaru-Utsumi (IU) local field correction function [21] local field correction function, Farid et al. (F) [22] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎤⎪ ⎡ ⎛q⎞ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛q⎞ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎢⎣ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜ ⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(9)

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Based on equations (8-9), Sarkar et al. (S) [23] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [21-23]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

− Z e2 W (r ) = rC3 =

−Z e r

⎡ ⎛ r ⎞⎤ 2 ⎢2 − exp ⎜⎜1 − ⎟⎟⎥ r ; ⎢⎣ ⎝ rC ⎠⎥⎦

2

;

r ≤ rC .

r ≥ rC

(11)

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

19

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is assumed that the potential within the core should not be zero nor constant but it should very as a function of r . Thus, the model potential has the novel feature of representing varying cancellation of potential within the core over and above its continuity at r = rC and weak nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows :

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ( ) Z ⎣ ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 24]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ 10 ⎠ ⎝ N ⎠ ⎝

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

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⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

20

Aditya M. Vora

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1. Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass Pd77.5Si16.5Cu

Z 3.05

ZS

Zd

1.50

7.16

Ω0 (au)

3

103.84

M (amu)

N

90.91

10.02

ρM (gm/cm3) 9.8073

rC (au) 0.7071

rd (au) 1.14

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 25],

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [14], Takeno-Goda (TG) [15, 16] and Bhatia-Singh (BS) [17, 18] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and BS approaches are given below [14-18]. According to the Hubbard-Beeby (HB) [14], the expressions for longitudinal and transverse phonon frequencies are as follows,

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ + ω2L (q ) = ω2E ⎢1 − − 2 σ q ( ) (qσ)3 ⎥⎦ σ q ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + 2 ( ) (qσ)3 ⎥⎦ σ q ⎣

(21)

∞

⎛ 4πρ ⎞ 2 with ω = ⎜ ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0 2 E

Following to Takeno-Goda (TG) [15, 16], the wave vector (q ) dependent longitudinal

and transverse phonon frequencies are written as

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

21

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎠⎭ ⎝ M ⎠0 ⎢⎣⎩ ⎝

{

ω L2 (q ) = ⎜

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ , (22) − + 2 3 ⎜3 ⎟ qr ( ) ( ) qr qr ⎝ ⎠⎥⎦

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − M qr ⎝ ⎠0 ⎝ ⎠⎭ ⎣⎢ ⎩

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . (23) ⎟⎥ rV ′(r )} ⎜⎜ + + 2 3 ⎟ 3 ( ) ( ) qr qr ⎝ ⎠ ⎦⎥

According to modified Bhatia-Singh (BS) [17, 18] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18],

k k 2 q 2ε (q ) G (qrS ) 2N ω (q ) = C2 (β I 0 + δ I 2 ) + e TF 2 2 ρq q + kTF ε (q ) 2 L

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ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ ρq ⎝ 2 ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [17, 18]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in earlier papers [1-18]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of

22

Aditya M. Vora

rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature θ D are found using the expressions [1-10],

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

θD

⎛ υ2 ⎞ 1 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = 2π ⎢ ⎥ = = kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) `

(31)

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [26]

CV =

Ω0 = 2 kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎤ ⎞ ⎛ ( ) = ω q λ λ ⎟⎟⎥ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎢exp⎜⎜ k T k B T ⎠⎦ ⎝ ⎠ ⎦⎣ ⎣ ⎝ B

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

23

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. Pd77.5Si16.5Cu6 0.51

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.41

V(r) (Ryd.)

0.31

0.21

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0.11

0.01 0

5

10

15

20

-0.09

r (au) Figure 1. Dependence on screening on pair potentials of Pd77.5Si16.5Cu6 bulk metallic glass.

The presently computed interatomic pair potentials are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the nature of the interatomic pair potential, significantly. The first zero of the interatomic pair potential V (r = r0 ) due to all local field correction functions occurs at r0 = 3.3 au. The interatomic pair potential well width and its minimum position Vmin (r ) are also affected by

the nature of the screening. The maximum depth in the interatomic pair potential is obtained for S-function. The present results do not show oscillatory behaviour and potential energy remains negative and constant in the large r - region. Thus, the Coulomb repulsive potential

24

Aditya M. Vora

part dominates the oscillations due to ion-electron-ion interactions, which show the waving shape of the interatomic pair potential after r = 10.0 au. Hence, the interatomic pair potentials converge towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the s- and d-electrons. Bretonnet and Derouiche [27] are observed that the repulsive part of V (r ) is drawn lower and its attractive part is deeper due to the d-electron effect and the V (r ) is shifted towards the lower r -values. Therefore,

the present results are supported the d-electron effect as noted by Bretonnet and Derouiche [27].

Pd77.5Si16.5Cu6 1.5

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

g(r)

1

0.5

0 0

5

10

-0.5

15

20

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-1

r (au) Figure 2. Dependence on screening on pair correlation function of Pd77.5Si16.5Cu6 bulk metallic glass.

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

2

25

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

HB 1.8

1.6

-1

ωL and ωT (in 10 Sec )

1.4

13

1.2

1

0.8

0.6

0.4

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0.2

Pd77.5Si16.5Cu6 0 0

1

2

q (A

0-1

3

4

5

)

Figure 3. Dependence on screening on phonon dispersion curves of Pd77.5Si16.5Cu6 bulk metallic glass using HB approach.

The pair correlation function (PCF) g (r ) computed theoretically through the interatomic pair potential is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.37, 1.55, 1.52, 1.52 and 1.60 for H-, T-, IU-, F- and S-function, respectively. While the ratio (r3 r1 ) of the position of the third peak (r3) to that of the first peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 1.89, 2.13, 2.09, 2.09 and 2.21 for H-, T-, IU-, F- and S-function, respectively. The (r2 r1 ) ratio is close to the c/a ratio in close-packed hexagonal structure i.e. c/a = 1.63, which

26

Aditya M. Vora

suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 9.4 au. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after r = 9.4 au in the Figure 3, because the experimental data is not available of this glass. Actually, this long range order is normal and it may be due to the waving shape of the interatomic pair potential. 1.2

TG

Pd77.5Si16.5Cu6

0.8

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

13

-1

ωL and ωT (in 10 Sec )

1

0.6

0.4

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.2

0 0

1

2

q (A

0-1

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Pd77.5Si16.5Cu6 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

27

5

BS

Pd77.5Si16.5Cu6

-1

ωL and ωT (in 10 Sec )

4

13

3

2

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Pd77.5Si16.5Cu6 bulk metallic glass using BS approach.

The experimental data of PDC of this glass is not available in the literature. Therefore, we have reported vibrational properties of this glass using pseudopotential theory for the first time. The phonon eigen frequencies for longitudinal and transverse phonon modes calculated using HB, TG and BS approaches with the five screening functions are shown in Figures 3-5. It can be seen that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present

28

Aditya M. Vora

results of PDC due to T-, IU- and F-function are lying between those due to H- and Sscreening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.2Å-1 for H-, q ≈ 3.2Å-1 for T-, IU-, F-function and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.4Å-1 for H-,

q ≈ 1.8Å-1 for T-, q ≈ 3.4Å-1 for IU- as well as F-function and q ≈ 1.9Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.5Å-1 for H-, T-, IU-, F- and S-function. Characteristically, the dispersion relations show a minimum near q p , the wave vector where the static structure factor S (q ) of the glass has its first maximum. The first maximum in the longitudinal branch found around at q ≈ 1.6Å

-1

ωL

of HB approach is

for H-, T-, IU- as well as F-function and q ≈ 1.7Å-1 for S-

function. While, the first maximum in the longitudinal branch of TG approach is found around at q ≈ 2.0Å-1 for H-, q ≈ 0.7Å-1 for T-, q ≈ 1.1Å-1 for IU- as well as F-function and

q ≈ 0.8Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.7Å-1 for most of the local field correction functions. It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon modes due to HB as well as TG approaches are more enhanced than the BS approach. The first crossover position of BS approaches is observed at q ≈ 2.1Å

-1

and q ≈ 1.5Å

-1

ωL

and

for most of the local field

correction functions, respectively. While, the first crossover position of Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ωT in the HB and

ωL

and

ωT in the

TG approach is observed at q ≈ 2.0Å for H-, q ≈ 1.8Å for T-, q ≈ 2.5Å for IU- as well -1

-1

-1

as F-function and q ≈ 1.4Å-1 for S-function. Here in transverse branch, the frequencies increase with the wave vector q and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [28] in which, it 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 Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) . The reason behind the anomalous behaviour may be due to 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 specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches.

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

29

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

HB 25

2

CV/T (10 J/(mol-K ))

20

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-2

15

10

5

Pd77.5Si16.5Cu6 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Pd77.5Si16.5Cu6 bulk metallic glass using HB approach.

30

Aditya M. Vora

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

TG

6

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-2

2

CV/T (10 J/(mol-K ))

8

4

2

Pd77.5Si16.5Cu6 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Pd77.5Si16.5Cu6 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

0.35

31

BS

0.3

2

CV/T (10 J/(mol-K ))

0.25

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-2

0.2

0.15

0.1 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.05

Pd77.5Si16.5Cu6 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Pd77.5Si16.5Cu6 bulk metallic glass using BS approach.

32

Aditya M. Vora

Table 2. Thermodynamic and Elastic properties of Pd77.5Si16.5Cu6 bulk metallic glass

υL x App.

HB

TG

BS

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Others [26]

SCR H T IU F S H T IU F S H T IU F S

5

υT x 5

10 cm/s 1.38 1.21 0.90 0.91 1.90 2.36 2.53 2.59 2.63 2.26 5.80 5.89 5.85 5.85 5.93

10 cm/s 0.80 0.70 0.52 0.52 1.10 1.11 1.46 1.42 1.45 1.31 1.89 2.03 1.97 1.98 2.06

-

-

BT x 1011 dyne/cm2 1.04 0.80 0.44 0.45 1.97 3.82 3.51 3.94 4.03 2.77 28.35 28.70 28.47 28.45 28.89 7.62, 9.06, 13.06, 13.66, 17.26, 18.30

G x 1011 2

dyne/cm

σ

0.62 0.48 0.27 0.27 1.18 1.21 2.09 1.96 2.06 1.69 3.52 4.03 3.82 3.83 4.16

0.25 0.25 0.25 0.25 0.25 0.36 0.25 0.29 0.28 0.25 0.44 0.43 0.44 0.44 0.43

3.39, 3.40

-

Y x 1011

θD

dyne/cm2

(K)

1.56 1.20 0.66 0.67 2.95 3.29 5.23 5.05 5.28 4.21 10.14 11.54 10.97 11.00 11.91 8.88, 9.06, 9.38, 9.41, 9.57, 9.60

105.90 92.87 69.17 69.75 145.77 149.92 194.12 189.00 193.41 174.38 258.25 275.94 268.90 269.24 280.40 303.60, 305.69, 309.44, 309.78, 311.67, 312.05

Furthermore, the thermodynamic and elastic properties estimated from the elastic part of the PDC are tabulated in Table 2. Among the five screening functions, the results of υ L and

υT are influenced more due to S-function. The comparison with other such theoretical results [26] favours the present calculation and suggests that proper choice of dielectric screening is important part for explaining the thermodynamic and elastic properties of Pd77.5Si16.5Cu6 glass. The results due to HB and TG approaches are lower than these due to BS approach. 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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While TG approach is developed upon the quasi-crystalline approximation in which effective force constant depends upon the correlation function for the

Study of Vibrational Dynamics of Pd77.5Si16.5Cu6 Bulk Metallic Glass

33

displacement of atoms and correlation function of displacement itself depends on the phonon frequencies. The BS approach is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties. From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study. The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

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CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

REFERENCES [1] [2]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. [3] Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. [4] Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. [5] Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. [6] Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. [7] Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. [8] Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. [9] Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. [10] Vora Aditya M. Romanian J. Phys. 2008, Vol. , pp - (in press).

34

Aditya M. Vora

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[11] Agarwal P. C.; Aziz K. A.; Kachhava C. M. Acta Phys. Hung. 1992, Vol. 72, pp 183-192. [12] Agarwal P. C.; Aziz K. A.; Kachhava C. M. Phys. Stat. Sol. (b). 1993, Vol. 178, pp 303-310. [13] Aziz K. A.; Ray A. K. J. Mater. Sci.: Mater. Elec. 1997, Vol. 8, pp 7-8. [14] Hubbard J.; Beeby J. L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. [15] Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352. [16] Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. [17] Bhatia A. B.; Singh R. N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. [18] Shukla M. M.; Campanha J. R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. [19] Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999. [20] Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. [21] Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. [22] Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621. [23] Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. [24] Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373. [25] Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. [26] Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. [27] Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. [28] Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 35-44 © 2011 Nova Science Publishers, Inc.

Chapter 3

STUDY OF VIBRATIONAL DYNAMICS OF BINARY MG70ZN30 METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, 370 001, Gujarat, India

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ABSTRACT The vibrational dynamics of Mg70Zn30 metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties. PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The homovalent Mg70Zn30 glass is one of the most important candidates of simple metallic glasses. The dynamical properties of Mg70Zn30 glass have been studied theoretically by von Heimendl [1] using the equation of motion method, by Tomenek [2] using a model calculation, by Saxena et al. [3] using effective pair potential and Takeno-Goda (TG) ∗

E-mail address: [email protected]. Tel. : +91-2832-256424.

36

Aditya M. Vora

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

approach [4], by Agarwal et al. [5] using Bhatia-Singh (BS) approach [6] and AgarwalKachhava [7] using TG as well as BS approaches. Experimentally, the phonon dispersion curves (PDC) for Mg70Zn30 glass was determined by Suck et al [8] for a few wave vector transfers near qP = 2.61Å-1, at which the first peak is found in static structure factor calculation. The atomic and electronic structure has been studied by Hafner-Jaswal [9] and Hafner et al [10] using ab initio pseudopotential technique. Benmore et al [11, 12] have been calculated longitudinal excitations within the first pseudo–Brillouin zone using the neutron Brillouin technique at room temperature. The temperature dependence of the dispersion and damping coefficients of transverse excitations was studied by Bryk and Mryglod [13] using the method of generalized collective modes. Thakore et al. [14] and Vora et al. [15, 16] have also been studied the PDC of Mg70Zn30 glass. In most of the above studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in binary alloys and metallic glasses [15-19]. Hence, in the present article the PAA model is used to investigate the phonon dynamics of A1XBX binary glassy system. The three theoretical approaches proposed by Hubbard-Beeby (HB) [20], Takeno-Goda (TG) [4] and Bhatia-Singh (BS) [6, 21] are used to generate the phonon dispersion curve (PDC). The local field correction function due to Sarkar et al [22] is employed for the first time to include the exchange and correlation effects in such study. Long wave-length limits of the phonon modes are then used to investigate the thermodynamic and elastic properties viz. isothermal bulk modulus (BT), modulus of rigidity (G), Young’s modulus (Y), longitudinal sound velocity (υL), transverse sound velocity (υT) and Debye temperature (θD).

2. THEORETICAL METHODOLOGY The pair potential V(r) is calculated from the relation given by Vora et al. [15, 16],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟⎟ + 2 ∫ F (q ) ⎢ V (r ) = ⎜⎜ ⎥ q dq . ⎣ qr ⎦ ⎝ r ⎠ π

(1)

Where, Z and ΩO are the valence and atomic volume of the glassy alloys, respectively. The energy wave number characteristics appearing in the Equation (1) is written as [15, 16],

[ε H (q )− 1] − ΩO q 2 2 F (q ) = WB (q ) . {1 + [ε H (q )− 1][1− f (q )]} 16 π

(2)

Here W B (q ) , ε H (q ) , f (q ) are the bare ion potential, the Hartree dielectric response function and the local field correction function to introduce the exchange and correlation effect, respectively.

Study of Vibrational Dynamics of Binary Mg70Zn30 Metallic Glass

37

The well recognized model potential W B (q ) used in the present computation of phonon dynamics of binary metallic glasses is of the form [15-19]. ⎡⎧ ⎤ U2 12 6U 2 18U 2 6U 4 ⎫ + + − ⎢⎪− 1 + 2 + ⎥ 3 2 2 2 3 ⎪ 2 2 + U U 1 1+U 1+U 1+U ⎪ ⎢⎪ ⎥ ⎬ cos (U )⎥ ⎢⎨ 2 4 ⎪ ⎢ ⎪ + 24 U 4 − 24U 4 ⎥ 2 ⎪ ⎢ ⎪⎩ 1 + U 2 ⎥ 1+U ⎭ ⎢ ⎥ + ⎢ ⎥ ⎢ ⎥ 3 ⎫ ⎧ U U U U 6 12 3 3 6 ⎢ ⎥ − + + − + ⎪ 2 2 3 ⎪ 3 ⎥ 1+U 2 1+U 2 1+U 2 1+U 2 ⎪ − 4 πe 2 Z ⎢ ⎪U U ⎢⎨ W B (q ) = ⎬ sin (U ) ⎥ 2 3 3 5 ΩO q 18U 6U 36U 6U ⎢⎪ ⎥ ⎪ + − + ⎢⎪− ⎥ 2 3 2 4 2 4 2 4 ⎪ 1+U 1+U 1+U ⎭ ⎢⎩ 1 + U ⎥ ⎢ ⎥ + ⎢ ⎥ 2 ⎧ ⎫ ⎢ ⎥ − U 1 ⎪ ⎪ 2 ⎢ 24 U exp (1) ⎨ ⎥ ⎬ 2 4 ⎢ ⎥ ⎩⎪ 1 + U ⎭⎪ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ .

(

(

) (

) ( (

) (

)

) (

) (

)

)

(

(

) (

) (

) (

)

)

(3)

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here U=qrc. This form has 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 [15-19]. Here the parameter rc is adjusted such that the calculated values of g(r) agree with the experimental value of g(r) as close as possible. A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [14, 15],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(4)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The three approaches for studying of phonons in amorphous alloys proposed by Hubbard -Beeby (HB) [20], Takeno-Goda (TG) [4] and Bhatia-Singh (BS) [6, 21] have been employed for studying the longitudinal and transverse phonon frequencies in three Mg-based glasses. According to the HB, the expressions for longitudinal and transverse phonon frequencies are as follows [20],

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ + − ω2L (q ) = ω2E ⎢1 − qσ (qσ)2 (qσ)3 ⎥⎦ ⎣

(5)

38

Aditya M. Vora

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ + ωT2 (q ) = ω2E ⎢1 − 2 ( ) (qσ)3 ⎥⎦ q σ ⎣ with

(6)

∞

⎛ 4πρ ⎞ 2 ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. 3 M ⎝ ⎠0

ω E2 = ⎜

Following to TG [4], the wave vector (q ) dependent longitudinal and transverse phonon frequencies are written as ∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎠⎭ ⎝ M ⎠0 ⎢⎣⎩ ⎝

{

ω L2 (q ) = ⎜

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ , − + ⎜3 qr (qr )2 (qr )3 ⎟⎠⎥⎦ ⎝

(7)

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − qr ⎝ M ⎠0 ⎠⎭ ⎝ ⎣⎢ ⎩

{

ω T2 (q ) = ⎜

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⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . ⎟⎥ rV ′(r )} ⎜⎜ + + 2 3 ⎟ 3 ( ) ( ) qr qr ⎝ ⎠ ⎥⎦

(8)

According to modified BS approach [6, 21], the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [21],

k k 2 q 2 G (qrS ) 2N ω (q ) = C2 (β I 0 + δ I 2 ) + e 2TF 2 ρq q + kTF ε (q ) 2 L

ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(9)

(10)

Other details of the constants used in this approach were already narrated in literature [6, 21]. Here M , ρ are the atomic mass and the number density of the glassy component while

V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

Study of Vibrational Dynamics of Binary Mg70Zn30 Metallic Glass

∴ ωL = υ L q

and ωT = υT q .

39 (11)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. For the three approaches the equations are : For HB approach the formulations for υL and υT are given as [20]

υL (HB) = ωE

3 σ2 , 10

(12)

υT (HB) = ωE

σ2 . 10

(13)

and

In TG approach the expressions for υL and υT are written as [4]

⎡ ⎛ 4π ρ ⎞ ⎟⎟ υL (TG) = ⎢ ⎜⎜ 30 M ⎠ ⎣⎝

∞

∫ 0

12

⎤ dr g(r ) r { r V ′′(r ) − 4V ′ (r )}⎥ , ⎦ 3

(14)

and

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⎡ ⎛ 4π ρ ⎞ ⎟⎟ υT (TG) = ⎢ ⎜⎜ 30 M ⎠ ⎣⎝

∞

∫ 0

12

⎤ dr g(r) r {3rV′′(r) − 4V′ (r)}⎥ ⎦ 3

.

(15)

The formulations for υL and υ T in BS approach are as follows [6, 21], 12

⎡N ⎛1 1 ⎞ k ⎤ υ L (BS ) = ⎢ C ⎜ β + δ ⎟ + e ⎥ , 5 ⎠ 3⎦ ⎣ ρ ⎝3

(16)

and

⎡N ⎛1 1 υ T (BS ) = ⎢ C ⎜ β + δ 15 ⎣ ρ ⎝3

12

⎞⎤ ⎟⎥ , ⎠⎦

(17)

In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature θ D are found using the expressions [15, 16],

40

Aditya M. Vora

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(18)

G = ρ υT2 .

(19)

ρ is the isotropic number density of the solid.

θD

⎛ υT2 ⎞ 1 − 2⎜⎜ 2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υT2 ⎞ 2 − 2⎜⎜ 2 ⎟⎟ ⎝ υL ⎠

(20)

Y = 2G (σ + 1) ,

(21)

⎡9 ρ ⎤ = ωD = = = 2π ⎢ ⎥ kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (22)

here ωD is the Debye frequency.

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3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are Z = 2.00, ΩO = 2.0472 x 10-23 cm3, rC = 4.8665 x 10-9 cm, NC =12.00, M = 5.871 x 10-23 gm and ρ = 2.9699 gm/cm3. The comparison of presently computed pair potential of Mg70Zn30 metallic glass and other such theoretical results [3, 7] are displayed in Figure 1. From the figure 1, it is seen that the first zero for V (r = r0 ) due to Sarkar et al’s (S) local field correction function occurs at

r0 = 5.8 au. The well width and the Vmin (r ) position of the pair potential are also affected by

the behaviour of the screening. It is also noticed that the well depth of presently computed pair potential is moved towards lower r-values and also shows lower as depth compared to the other theoretical results [3, 7]. The presently computed results of the pair potential are found in qualitatively good agreement with the theoretical reported data [3, 7]. The pair potential V(r) of Saxena et al [3] and Agarwal et al [7] are highly oscillatory. Such oscillation for large r-region is not present in the computation and V(r) converges very rapidly to zero for higher r-values. The result of Saxena et al [3] and Agarwal et al [7] shows significant oscillations and potential energy remains positive in the large r–region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions in their studies.

Study of Vibrational Dynamics of Binary Mg70Zn30 Metallic Glass

41

[3] [7]

Figure 1. Pair Potential of Mg70Zn30 metallic glass.

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[7]

Figure 2. Pair Correlation Function (g(r)) of Mg70Zn30 metallic glass.

The presently computed g(r) of Mg70Zn30 metallic glass is displayed in Fig. 2 alongwith experimental [11] and other such theoretical data [7]. An excellent agreement of presently computed g(r) with experimental data [11] is seen around the first peak. The structural information obtained from the g(r) of Mg70Zn30 metallic glass is listed in Table 1. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) for is close to the c/a ratio in close-packed hexagonal structure i.e. c/a = 1.59. This means that the short range

42

Aditya M. Vora

order of near neighbour in the amorphous state is affected more or less by the atomic arrangement of crystalline state. The ratio (r3 r1 ) of the position of the third peak (r3) to that of the third peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is also calculated and found 2.14. Table 1. Peak poisons and peak ratios in the pair correlation functions of Mg70Zn30 metallic glass Structural data First peak position r1 (a.u.) Second peak position r2 (a.u.) Third peak position r3 (a.u.) Ratio r2/r1 Ratio r3/r1

Present Results 5.8 9.2 12.4 1.59 2.14

[8]

[1]

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[3] [7]

Figure 3. Phonon Dispersion Curves of Mg70Zn30 metallic glass.

The presently calculated PDC due to three approaches are shown in Fig. 3 alongwith available theoretical [1, 3, 7] and experimental data [8]. The longitudinal branch of PDC calculated from BS approach is higher than those obtained by other two theoretical approaches. The presently computed values of the PDC form HB and TG approaches show lower results in comparison with the other reported vales. First minima in the longitudinal branch lie near to q P value at which the structure factor S(q ) shows its first peak [11]. It is noticed from the Fig. 3 that, the first minimum in the longitudinal branch is seen around at

Study of Vibrational Dynamics of Binary Mg70Zn30 Metallic Glass

43

q ≈ 1.8Å-1 for BS, q ≈ 2.5Å-1 for TG and HB approaches, respectively. The first crossing position of

ω L and ω T branches is observed at 2.0Å-1 in HB, 1.9Å-1 in TG and 1.6Å-1 in BS

approach. In comparison to the other reported data [1, 3, 7, 8], the present results are suppressed one. Table 2. Thermodynamic and Elastic Properties of Mg70Zn30 metallic glass Prop. υL x 105 cm / sec υT x 105 cm / sec BT x 1011 dyne / cm2 G x 1011 dyne / cm2 Y x 1011 dyne / cm2 θD K

HB 1.5590 1.0829 0.2671 0.3612 0.7468 129.74

TG 1.4140 1.0513 0.1619 0.3404 0.6005 124.62

BS 6.0506 1.9760 9.6723 1.2026 3.8464 247.86

Others [3] 4.7, 5.1 2.5, 2.6 – – – 305.21

The computed thermodynamics properties are listed in Table 2. The results due to HB and TG approaches are lower than these due to BS approach. As experimental data for these properties are not available in the literature, hence it is very difficult to put further comments on it. Here, longitudinal and transverse sound velocities due to BS approach show good agreement with other such theoretical outcome [3]. The comparison with other such results [3] favours the present calculation and suggests that proper choice of dielectric screening is important part for explaining the thermodynamic and elastic properties of Mg70Zn30 glass.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches reproduce all the broad characteristics of dispersion curves. The well recognized model potential with more advanced S–local field correction function generates consistent results. Hence, the model potential is suitable for studying the vibrational dynamics of Mg70Zn30 metallic glass, which confirms not only the applicability of the model potential in the aforesaid properties but also supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and metallic glasses is in progress.

REFERENCES [1] [2] [3] [4] [5] [6]

L. von Heimendahl, J Phys F: Met. Phy 9 (1979) 161. D. Tomanek, Diplomarbeit, Universität Basel (1979). N. S. Saxena, Deepika Bhandari, Arun Pratap, M. P. Saksena, J Phys: Condens Matter 2 (1990) 9475. S. Takeno, M. Goda, Prog. Thero. Phys. 45 (1971) 331; Prog. Thero. Phys. 47 (1972) 790. P. C. Agarwal, K. A. Aziz, C. M. Kachhava, Acta Phys Hung. 72 (1992) 183; Phys.Stat Sol (b) 178 (1993) 303. A. B. Bhatia, R. N. Singh, Phys Rev. B31 (1985) 4751.

44 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

[17] [18] [19]

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[20] [21] [22]

Aditya M. Vora P. C. Agarwal, C. M. Kachhava, Physica B179 (1992) 43; Phys. Stat. Sol. (b) 179 (1993) 365. J. –B. Suck, H. Rudin, H. –J. Güntherodt, H. Beck, J. Phys. C: Solid State Phys. 13 (1980) L1045. J. Hafner, S. S. Jaswal, Phil. Mag. A58 (1988) 61. J. Hafner, S. S. Jaswal, M. Tegze, A. Pflugi, J. Krieg, P. Oelhafen, H. –J. Güntherodt, J. Phys. F: Met. Phys. 18 (1988) 2583. C. J. Benmore, B. J. Oliver, J. –B. Suck, R. A. Robinsonk, P. A. Eglestaff, J. Phys.: Condens. Matter 7 (1995) 4775. C. J. Benmore, S. Sweeney, R. A. Robinson, P. A. Eglestaff, J. –B. Suck, Physica B241-243 (1998) 912; J. Phys.: Condens. Matter 11 (1999) 7079. T. Bryk, I. Mryglod, Cond. Mat. Phys. 2 (1999) 285. B. Y. Thakore, P. N. Gajjar, A. R. Jani, Solid State Physics (India) 40C (1997) 70; Bull. Mater. Sci. 23 (2000) 5. A. M. Vora, M. H. Patel, P. N. Gajjar, A. R. Jani, Solid State Physics (India) 46 (2003) 315. Aditya M Vora, Chinese Phys. Lett. 23 (2006) 1872; J. Non-Cryst. Sol. 352 (2006) 3217; J. Mater. Sci. 43 (2007) 935; Acta Phys. Polo. A111 (2007) 859; Front. Mater. Sci. China 1 (2007) 366, Fizika A16 (2007) 187. M. H Patel, A. M. Vora, P. N. Gajjar, A. R. Jani, Physica B304 (2001) 152; Commun. Theor. Phys. 38 (2002) 365. P. N. Gajjar, A. M. Vora, M. H. Patel, A. R. Jani, Int. J. Mod. Phys. B17 (2003) 6001. P. N. Gajjar, A. M. Vora, A. R. Jani, Mod. Phys. Lett. B18 (2004) 573; Ind. J. Phys. 78 (2004) 775. J. Hubbard, J. L. Beeby, J. Phys. C: Solid State Phys. 2 (1969) 556. M. M. Shukla, J. R. Campanha, Acta Phys. Pol. A94 (1998) 655. A. Sarkar, D. S. Sen, S. Haldar, D. Roy, Mod. Phys. Lett. B12 (1998) 639.

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 45-63 © 2011 Nova Science Publishers, Inc.

Chapter 4

STUDY OF VIBRATIONAL DYNAMICS OF FE80B14SI6 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, 370 001, Gujarat, India

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ABSTRACT The vibrational dynamics of Fe80B14Si6 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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∗

E-mail address : [email protected], Tel. : +91-2832-256424.

46

Aditya M. Vora

16]. The Fe80B14Si6 metallic glass is the most important candidate of transition metalmetalloid group. In our literature survey we have not found any reports on the experimental as well as theoretical work based on pseudopotential theory related to PDC of this glass. But, the structural properties of this glass have been studied by Sulir et al. [11]. Therefore, the vibrational properties of this glass are reported for the first time. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio

σ

and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 22],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ F (q ) ⎢ Vs (r ) = ⎜ S ⎥ q dq . ⎜ r ⎟ π2 ∫ ⎣ qr ⎦ ⎝ ⎠

(2)

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

47

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [22], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2

16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [17]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [17],

ε H (q ) = 1 +

m e2 2 π k = 2 η2 F

⎛ 1 − η2 ⎞ 1+ η ⎜ ⎟ ;η = q ln + 1 ⎜ 2η ⎟ 1− η 2k F ⎝ ⎠

(5)

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [17] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [18] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [18],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ ⎢1 + ⎥. π k F2 ⎦ ⎣

(7)

The Ichimaru-Utsumi (IU) local field correction function [19] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities,

48

Aditya M. Vora

which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎛ ⎤⎪ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ . f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛ q ⎞⎪ ⎢⎣ ⎝ kF ⎠ ⎝ ⎥⎦ ⎪ ⎛ q ⎞ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ 4⎜ ⎟ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(8) On the basis of Ichimaru-Utsumi (IU) local field correction function [19] local field correction function, Farid et al. (F) [20] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎤⎪ ⎡ ⎛q⎞ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎢⎣ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 4 2 − ⎜⎜ ⎟⎟ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(9)

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Based on equations (8-9), Sarkar et al. (S) [21] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [19-21]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

W (r ) =

− Z e2 rC3

− Z e2 = r

⎡ ⎛ r ⎞⎤ 2 ⎢2 − exp ⎜⎜1 − ⎟⎟⎥ r ; ⎢⎣ ⎝ rC ⎠⎥⎦ ;

r ≤ rC .

(11)

r ≥ rC

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

49

assumed that the potential within the core should not be zero nor constant but it should very as a function of r . Thus, the model potential has the novel feature of representing varying cancellation of potential within the core over and above its continuity at r = rC and weak nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows :

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ⎣ (Z ) ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 22]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ ⎝ 10 ⎠ ⎝ N ⎠

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

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Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from

50

Aditya M. Vora

the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1. Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass Fe80B14Si6

Z 3.06

ZS

Zd

1.50

5.20

Ω0 3

M (amu)

N

77.52

47.88

6.40

(au)

ρM (gm/cm3)

rC (au)

(au)

6.9190

0.6400

1.21

rd

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 23],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

BS approaches are given below [12-16]. According to the Hubbard-Beeby (HB) [12], the expressions for longitudinal and transverse phonon frequencies are as follows,

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ ω2L (q ) = ω2E ⎢1 − − + 2 σ q ( ) (qσ)3 ⎥⎦ σ q ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + (qσ)2 (qσ)3 ⎥⎦ ⎣

(21)

∞

⎛ 4πρ ⎞ 2 with ω = ⎜ ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0 2 E

Following to Takeno-Goda (TG) [13, 14], the wave vector (q ) dependent longitudinal and transverse phonon frequencies are written as

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

51

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎠⎭ ⎝ M ⎠0 ⎢⎣⎩ ⎝

{

ω L2 (q ) = ⎜

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ , − + 2 3 ⎜3 ⎟ qr ( ) ( ) qr qr ⎝ ⎠⎥⎦

(22)

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − M qr ⎠0 ⎝ ⎠⎭ ⎝ ⎣⎢ ⎩

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . ⎟⎥ + rV ′(r )} ⎜⎜ + 2 3 ⎟ 3 ( ) ( ) qr qr ⎝ ⎠ ⎦⎥

(23)

According to modified Bhatia-Singh (BS) [15, 16] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18],

ωL2 (q ) =

2 ke kTF q 2ε (q ) G (qrS ) 2 NC ( ) I I β δ + + 0 2 2 ρ q2 q 2 + kTF ε (q )

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ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [15, 16]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in earlier papers [1-16]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of

52

Aditya M. Vora

rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature θ D are found using the expressions [1-10],

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

θD

⎛ υ2 ⎞ 1 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = 2π ⎢ ⎥ = = kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (31)

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here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [24]

CV =

Ω0 = 2 kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎤ ⎛ ⎞ ( ) = ω q λ λ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎟⎟⎥ ⎢exp⎜⎜ k T k B T ⎠⎦ ⎠ ⎦⎣ ⎝ ⎣ ⎝ B

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

53

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. 0.55

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Fe80B14Si6

0.35

V(r) (Ryd.)

0.15

-0.05 0

5

10

15

20

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

-0.25

-0.45

-0.65

r (au) Figure 1. Dependence on screening on pair potentials of Fe80B14Si6 bulk metallic glass.

The presently computed interatomic pair potentials of this glass are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the nature of the interatomic pair potential, significantly. The first zero of the interatomic pair potential V (r = r0 ) due to all local field correction functions occurs at

r0 = 2.3 au. The interatomic pair potential well width and its minimum position Vmin (r ) are

also affected by the nature of the screening. The maximum depth in the interatomic pair potential is obtained for F-function, while minimum is for H-screening function. The present results do not show oscillatory behaviour and potential energy remains negative in the large

54

Aditya M. Vora

r - region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ionelectron-ion interactions, which show the waving shape of the interatomic pair potential after r = 7.0 au. Hence, the interatomic pair potentials converge towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the sand d-electrons. Bretonnet and Derouiche [25] are observed that the repulsive part of V (r ) is

drawn lower and its attractive part is deeper due to the d-electron effect and the V (r ) is

shifted towards the lower r -values. Therefore, the present results are supported the delectron effect as noted by Bretonnet and Derouiche [25]. 4.5

HB

Fe80B14Si6

4

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13

-1

ωL and ωT (in 10 Sec )

3.5

3

2.5

2

1.5

1 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 2. Dependence on screening on pair correlation function of Fe80B14Si6 bulk metallic glass.

The pair correlation function (PCF) g (r ) computed theoretically through the interatomic pair potential is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.28, 1.51, 1.51, 1.51 and 1.57 for H-, T-, IU-, F- and S-function,

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

55

respectively. While the ratio (r3 r1 ) of the position of the third peak (r3) to that of the first peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 1.77, 2.08, 2.08, 2.08 and 2.16 for H-, T-, IU-, F- and S-function, respectively. The (r2 r1 ) ratio is close 1.63 characteristic for the disordered closed pack crystallographic structures and therefore this component of the second peak might eventually be due to not complete amorphousation of the samples [11]. This suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 8.5 au. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after r = 8.5 au in the Figure 2. Actually, this long range order is normal and it may be due to the waving shape of the interatomic pair potential. 1.5

Fe80B14Si6

1

g(r)

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0.5

0 0

5

10

15

20

-0.5 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-1

r (au)

Figure 3. Dependence on screening on phonon dispersion curves of Fe80B14Si6 bulk metallic glass using HB approach.

56

Aditya M. Vora 3

TG

Fe80B14Si6

2

13

-1

ωL and ωT (in 10 Sec )

2.5

1.5

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0 0

1

2

q (A

0-1

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Fe80B14Si6 bulk metallic glass using TG approach.

The phonon eigen frequencies for longitudinal and transverse phonon modes calculated using HB, TG and BS approaches with the five screening functions are shown in Figures 3-5. It can be seen that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present results of the PDC due to H-, T- and IU-function are lying between those due to F- and S-screening in HB approach. While, the computed outcomes of the PDC due to T-, IU- and F-function are lying between those due to H- and S-screening in TG approach. Also in BS approach, the present results of the PDC due to T-, IU- and F-function are lying between those due to H- and Sscreening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.5Å-1 for H-, T-, IU-, F- and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.9Å-1 for H-, q ≈ 3.7Å-1 for T-, q ≈ 3.2Å-1 for IU- as well as F-function and q ≈ 2.2Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.5Å-1 for H-, T-, IU-, F- and S-function. The

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass first maximum in the longitudinal branch

ωL

57

of HB approach is found around at q ≈ 1.8Å-1

for H-, T-, IU-, F- and S-function. While, the first maximum in the longitudinal branch of TG approach is found around at q ≈ 1.4Å-1 for H-, T-, IU- as well as F-function and q ≈ 0.9Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.7Å-1 for most of the local field correction functions. 8

BS

Fe80B14Si6

7

-1

ωL and ωT (in 10 Sec )

6

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13

5

4

3

2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Fe80B14Si6 bulk metallic glass using BS approach.

It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon

58

Aditya M. Vora

modes due to HB as well as TG approaches are more enhanced than the BS approach. The first crossover position of 1

ωL

and

ωT in the HB and BS approaches is observed at q ≈ 2.7Å-

and q ≈ 1.5Å-1 for most of the local field correction functions, respectively. While, the first

ωL

ωT in the TG approach is observed at q ≈ 2.2Å-1 for H-, q ≈ 2.6Å-1 for T-, q ≈ 2.5Å-1 for IU- as well as F-function and q ≈ 1.7Å-1 for S-function.

crossover position of

and

Here in transverse branch, the frequencies increase with the wave vector q and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [26] in which, it 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.

Fe80B14Si6

HB

30

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-2

2

CV/T (10 J/(mol-K ))

25

20

15

10

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

5

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Fe80B14Si6 bulk metallic glass using HB approach.

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

59

8

TG

Fe80B14Si6

-2

2

CV/T (10 J/(mol-K ))

6

4

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2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

20

40

60

2

2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Fe80B14Si6 bulk metallic glass using TG approach.

As shown in Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) . The reason behind the anomalous behaviour may be due to 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 specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches.

60

Aditya M. Vora 0.3

BS

0.25

-2

2

CV/T (10 J/(mol-K ))

0.2

0.15

0.1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0.05

Fe80B14Si6 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Fe80B14Si6 bulk metallic glass using BS approach.

It is noticed from Table 2 that the υ L and υT for HB and TG approaches are influenced significantly by various exchange and correlation functions as compare for BS approach. However, very high compressibility is observed for BS approach. As υ L and υT are depend on the nature of screening as well as the method adopted, the other thermodynamic and elastic properties are also reflecting the same behaviour.

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

61

Table 2. Thermodynamic and Elastic properties of Fe80B14Si6 bulk metallic glass App.

HB

TG

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BS

SCR H T IU F S H T IU F S H T IU F S

υL x 105 υT x 105 BT x 1011 cm/s 2.95 3.67 3.93 3.92 2.23 4.28 4.09 4.26 4.31 3.39 8.72 8.81 8.78 8.78 8.83

cm/s 1.70 2.12 2.27 2.26 1.29 2.05 2.27 2.28 2.31 1.94 2.61 2.75 2.71 2.71 2.78

2

dyne/cm 3.33 5.17 5.95 5.92 1.91 8.78 6.79 7.79 7.90 4.46 46.29 46.69 46.52 46.50 46.84

G x 1011 2

dyne/cm 2.00 3.10 3.57 3.55 1.14 2.92 3.57 3.58 3.70 2.62 4.70 5.24 5.08 5.09 5.36

σ 0.25 0.25 0.25 0.25 0.25 0.35 0.28 0.30 0.30 0.25 0.45 0.45 0.45 0.45 0.44

Y x 1011

θD

dyne/cm2

(K) 249.22 310.35 332.92 331.94 188.46 304.79 333.93 335.62 340.92 285.10 392.00 413.86 407.65 407.85 418.50

5.00 7.76 8.93 8.87 2.86 7.88 9.11 9.32 9.60 6.56 13.63 15.16 14.72 14.73 15.49

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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While 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 is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties. From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study. The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to

62

Aditya M. Vora

study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

REFERENCES

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. Vora Aditya M. Romanian J. Phys. 2008, Vol. 53, pp 517-533. Sulir L. D.; Pyka M.; Kozlowski A. Acta Phys. Pol. A. 1983, Vol. 63, pp 435-443. Hubbard J.; Beeby J. L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352. Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. Bhatia A. B.; Singh R. N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. Shukla M. M.; Campanha J. R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999. Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621. Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373.

Study of Vibrational Dynamics of Fe80B14Si6 Bulk Metallic Glass

63

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[23] Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. [24] Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. [25] Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. [26] Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

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In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 65-84 © 2011 Nova Science Publishers, Inc.

Chapter 5

STUDY OF VIBRATIONAL DYNAMICS OF FE80B10SI10 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, Gujarat, India

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ABSTRACT The vibrational dynamics of Fe80B10Si10 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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 [116]. The Fe80B10Si10 metallic glass is the most important candidate of transition metal∗

E-mail address: [email protected]; Tel.: +91-2832-256424.

66

Aditya M. Vora

metalloid group. In our literature survey we have not found any reports on the experimental as well as theoretical work based on pseudopotential theory related to PDC of this glass. But, the structural properties of this glass have been studied by Sulir et al. [11]. Therefore, the vibrational properties of this glass are reported for the first time. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ

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and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 22],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ F (q ) ⎢ Vs (r ) = ⎜ S ⎥ q dq . ⎜ r ⎟ π2 ∫ qr ⎦ ⎣ ⎝ ⎠

(2)

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

67

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [22], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2

16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [17]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [17],

ε H (q ) = 1 +

m e2 2 π k = 2 η2 F

⎞ ⎛ 1 − η2 1+ η ⎟ ;η = q ⎜ + 1 ln ⎟ ⎜ 2η 2k F 1− η ⎠ ⎝

(5)

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

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and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [17] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [18] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [18],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ . ⎢1 + 2 ⎥ π k F ⎦ ⎣

(7)

The Ichimaru-Utsumi (IU) local field correction function [19] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self

68

Aditya M. Vora

consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎤ . (8) k ⎛q⎞ ⎛ ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎪ ⎝ kF ⎠ ⎪ f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎝ ⎥⎦ ⎪ ⎛ q ⎞ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜ ⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩

On the basis of Ichimaru-Utsumi (IU) local field correction function [19] local field correction function, Farid et al. (F) [20] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . (9) f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜ ⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩ Based on equations (8-9), Sarkar et al. (S) [21] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

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The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [19-21]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

W (r ) = =

⎛ r ⎞⎤ 2 − Z e2 ⎡ ⎢2 − exp ⎜⎜1 − ⎟⎟⎥ r ; 3 rC ⎣⎢ ⎝ rC ⎠⎦⎥ −Z e r

2

;

r ≤ rC .

(11)

r ≥ rC

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is assumed that the potential within the core should not be zero nor constant but it should very as a function of r . Thus, the model potential has the novel feature of representing varying cancellation of potential within the core over and above its continuity at r = rC and weak

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

69

nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows:

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ⎣ (Z ) ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 22]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ ⎝ 10 ⎠ ⎝ N ⎠

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

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orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1.

70

Aditya M. Vora Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass

Z

ZS

Zd

Fe80B10Si10

3.10

1.50

5.20

Ω0 3

M (amu)

N

ρM (gm/cm3)

rC (au)

80.93

48.57

6.40

6.7230

0.6437

(au)

rd (au) 1.21

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 23],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and

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BS approaches are given below [12-16]. According to the Hubbard-Beeby (HB) [12], the expressions for longitudinal and transverse phonon frequencies are as follows,

with

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ ω2L (q ) = ω2E ⎢1 − − + qσ (qσ)2 (qσ)3 ⎥⎦ ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + (qσ)2 (qσ)3 ⎥⎦ ⎣

(21)

∞

⎛ 4πρ ⎞ 2 ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0

ω E2 = ⎜

Following to Takeno-Goda (TG) [13, 14], the wave vector (q ) dependent longitudinal and transverse phonon frequencies are written as ∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎝ M ⎠0 ⎝ ⎠⎭ ⎣⎢⎩

ω L2 (q ) = ⎜

{

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ − + 2 3 ⎜3 ⎟ qr ( ) ( ) qr qr ⎝ ⎠⎥⎦

,

(22)

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass ∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − qr ⎝ M ⎠0 ⎠⎭ ⎝ ⎣⎢ ⎩

71

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞⎤ ⎟⎥ rV ′(r )} ⎜⎜ + + (qr )2 (qr )3 ⎟⎠⎥⎦ ⎝3

.

(23)

According to modified Bhatia-Singh (BS) [15, 16] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18],

k k 2 q 2ε (q ) G (qrS ) 2N ω (q ) = C2 (β I 0 + δ I 2 ) + e TF 2 2 ρq q + kTF ε (q ) 2 L

ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [15, 16]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

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ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in the earlier papers [1-16]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature

θ D are found using the expressions [1-10],

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

72

Aditya M. Vora

θD

⎛ υ2 ⎞ 1 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = 2π ⎢ ⎥ = = kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (31)

here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [24]

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Ω0 = 2 CV = kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎛ = ω λ (q ) ⎞⎤ λ ⎟⎟⎥ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎢exp⎜⎜ k T k T B B ⎝ ⎠⎦ ⎝ ⎠ ⎣ ⎦⎣

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. The presently computed interatomic pair potentials are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the nature of the interatomic pair potential, significantly. The first zero of the interatomic pair potential V (r = r0 ) due to all local field correction functions occurs at r0 = 2.2 au. The interatomic pair potential well width and its minimum position Vmin (r ) are also affected by the

nature of the screening. The maximum depth in the interatomic pair potential is obtained for Ffunction. The present results do not show oscillatory behaviour and potential energy remains negative in the large r - region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions, which show the waving shape of the interatomic pair potential after r = 5.7 au. Hence, the interatomic pair potentials converge

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

73

towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the s- and d-electrons. Bretonnet and Derouiche [25] are observed that the repulsive part of V (r ) is drawn lower and its attractive part is deeper due to the d-

electron effect and the V (r ) is shifted towards the lower r -values. Therefore, the present results are supported the d-electron effect as noted by Bretonnet and Derouiche [25].

Fe80B10Si10

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.4

0.2

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V(r) (Ryd.)

0 0

5

10

15

20

-0.2

-0.4

-0.6

-0.8

r (au) Figure 1. Dependence on screening on pair potentials of Fe80B10Si10 bulk metallic glass.

74

Aditya M. Vora

1.5

Fe80B10Si10

1

g(r)

0.5

0 0

5

10

15

20

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

-0.5 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-1

r (au) Figure 2. Dependence on screening on pair correlation function of Fe80B10Si10 bulk metallic glass.

The pair correlation function (PCF) g (r ) computed theoretically through the interatomic pair potential is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.56, 1.53, 1.50, 1.53 and 1.59 for H-, T-, IU-, F- and S-function, respectively. While the ratio (r3 r1 ) of the position of the third peak (r3) to that of the first peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 2.14, 2.10, 2.06, 2.10 and 2.18 for H-, T-, IU-, F- and S-function, respectively. The

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

(r2 r1 )

75

ratio is close 1.63 characteristic for the disordered closed pack crystallographic

structures and therefore this component of the second peak might eventually be due to not complete amorphousation of the samples [11]. This suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 8.5 au. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after r = 8.5 au in the Figure 2. In fact, this long range order is normal and it may be due to the waving shape of the interatomic pair potential. The phonon eigen frequencies for longitudinal and transverse phonon modes calculated using HB, TG and BS approaches with the five screening functions are shown in Figures 3-5. It can be seen that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present results of the PDC due to H-, T- and F-function are lying between those due to IU- and S-screening in HB approach. While, the computed outcomes of the PDC due to T-, IU- and F-function are lying between those due to H- and S-screening in TG approach. Also in BS approach, the present results of the PDC due to T-, IU- and F-function are lying between those due to H- and Sscreening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.5Å-1 for H-, T-, IU-, F- and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.9Å-1 for H-, q ≈ 3.6Å-1 for T-, q ≈ 3.3Å-1 for IU- as well as F-function and q ≈ 2.2Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.5Å-1 for H-, T-, IU-, F- and S-function. The

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first maximum in the longitudinal branch

ωL

of HB approach is found around at q ≈ 1.8Å-1

for H-, T-, IU-, F- and S-function. While, the first maximum in the longitudinal branch of TG approach is found around at q ≈ 1.4Å-1 for H-, T-, IU- as well as F-function and q ≈ 0.9Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.7Å-1 for most of the local field correction functions. It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon modes due to HB as well as TG approaches are more enhanced than the BS approach. The first crossover position of 1

ωL

and

ωT in the HB and BS approaches is observed at q ≈ 2.7Å-

and q ≈ 1.5Å-1 for most of the local field correction functions, respectively. While, the first

ωL

ωT in the TG approach is observed at q ≈ 2.3Å-1 for H-, T-, IU- as well as F-function and q ≈ 1.7Å-1 for S-function. Here in transverse branch, the crossover position of

and

frequencies increase with the wave vector q and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [26] in which, it 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.

76

Aditya M. Vora

5

HB

Fe80B10Si10

4.5

4

-1

ωL and ωT (in 10 Sec )

3.5

13

3

2.5

2

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1.5

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

0-1

q (A

3

4

5

)

Figure 3. Dependence on screening on phonon dispersion curves of Fe80B10Si10 bulk metallic glass using HB approach.

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

77

3

TG

Fe80B10Si10

2

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13

-1

ωL and ωT (in 10 Sec )

2.5

1.5

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Fe80B10Si10 bulk metallic glass using TG approach.

78

Aditya M. Vora

8

BS

Fe80B10Si10

7

-1

ωL and ωT (in 10 Sec )

6

13

5

4

3

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2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Fe80B10Si10 bulk metallic glass using BS approach.

As shown in Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) . The reason behind the anomalous behaviour may be due to 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

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

79

.

specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches. It is apparent from the nature that CV is more sensitive to screening. The initial rise on the CV T values is observed for low temperature and then further increase of temperature give convergent value.

30

HB

20

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-2

2

CV/T (10 J/(mol-K ))

25

15

10

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

5

Fe80B10Si10 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Fe80B10Si10 bulk metallic glass using HB approach.

80

Aditya M. Vora

8 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

TG

-2

2

CV/T (10 J/(mol-K ))

6

4

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2

Fe80B10Si10 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Fe80B10Si10 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

81

BS 0.3

0.2

-2

2

CV/T (10 J/(mol-K ))

0.25

0.15

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0.1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.05

Fe80B10Si10 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Fe80B10Si10 bulk metallic glass using BS approach.

The calculated thermodynamic and elastic properties of this glass are shown in Table 2. Here it is found that the incorporation of exchange and correlation effects in the static Hdielectric function enhance the longitudinal and transverse sound velocities in HB and TG

82

Aditya M. Vora

approaches while in BS approach suppression on the velocities is observed. For this glass, also, we do not have any comparison for BT , G , σ , Y and θ D hence we avoid to put any remarks. Table 2. Thermodynamic and Elastic properties of Fe80B10Si10bulk metallic glass App

HB

TG

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BS

SCR H T IU F S H T IU F S H T IU F S

υL x 5

10 cm/s 3.08 3.83 4.12 4.10 2.32 4.35 4.09 4.30 4.34 3.41 8.54 8.62 8.59 8.59 8.64

υT x 5

10 cm/s 1.78 2.21 2.38 2.37 1.34 2.08 2.28 2.29 2.33 1.95 2.44 2.56 2.53 2.53 2.59

BT x 1011 2

dyne/cm 3.53 5.48 6.33 6.29 2.01 8.85 6.60 7.71 7.80 4.40 43.74 44.05 43.91 43.90 44.17

G x 1011 2

dyne/cm 2.12 3.29 3.80 3.77 1.21 2.90 3.50 3.53 3.64 2.56 3.99 4.42 4.29 4.29 4.50

σ 0.25 0.25 0.25 0.25 0.25 0.35 0.28 0.30 0.30 0.26 0.46 0.45 0.45 0.45 0.45

Y x 1011 dyne/cm2 5.30 8.22 9.50 9.44 3.02 7.84 8.91 9.19 9.46 6.43 11.61 12.82 12.46 12.48 13.07

θD (K) 256.56 319.42 343.44 342.33 193.62 303.95 330.40 333.12 338.28 281.93 361.42 380.07 374.60 374.92 383.87

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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While 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 is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties. From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study. The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static

Study of Vibrational Dynamics of Fe80B10Si10 Bulk Metallic Glass

83

dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

REFERENCES

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. Vora Aditya M. Romanian J. Phys. 2008, Vol. , pp - (in press). Sulir L. D.; Pyka M.; Kozlowski A. Acta Phys. Pol. A. 1983, Vol. 63, pp 435-443. Hubbard J.; Beeby J.L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352. Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. Bhatia A. B.; Singh R.N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. Shukla M. M.; Campanha J.R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999. Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621.

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[21] Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. [22] Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373. [23] Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. [24] Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. [25] Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. [26] Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 85-94 © 2011 Nova Science Publishers, Inc.

Chapter 6

COMPUTATION OF PHONON DISPERSION CURVES OF CA70MG30 METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, Gujarat, India

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ABSTRACT The computation of the phonon dispersion curves of Ca70Mg30 metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). Our recently proposed model potential is employed successfully to explain electron-ion interaction in the metallic glass. The local field correction function due to Sarkar et al is used for the first time to introduce the exchange and correlation effects in the aforesaid properties. The effective pair potential is used to generate the pair correlation function g(r). The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic and elastic properties viz. longitudinal and transverse sound velocities, isothermal bulk modulus, modulus of rigidity, Poisson’s ratio, Young’s modulus and Debye temperature are also investigated successfully.

Keywords: Pseudopotential, metallic glass, phonon dispersion curves, thermal properties, elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION Ca70Mg30 glass is the most important candidate of simple metallic glasses. The phonon dispersion curves (PDC) of this glass have been investigated by many workers using the pseudopotential theory [1-8]. The experimental observation was made by Suck et al. [9] The theoretical investigations of Hafner [1] and Hafner and Jaswal [2] on Ca70Mg30 glass are on ∗

Tel. : +91-2832-256424, E-mail address : [email protected].

86

Aditya M. Vora

the basis of S (q, ω ) and by Bhatia-Singh (BS) approach [10] assuming the force among nearest neighbours as central and volume dependent. Saxena et al [3, 4] have studied the PDC of this glass using the Hubbard-Beeby (HB) [11] and Takeno-Goda (TG) [12] approaches with effective pair potential (EPP) and effective atom (EAM) models. Agarwal et al [5] and Agarwal and Kachhava [6] have calculated the PDC of the glass using BS approach. Shukla and Campanha [7] have studied PDC of this glass using modified BS approach. Thakore et al [8] have studied the PDC and their related properties of the glass using HB approach with EAM model. In most of the above studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is good in agreement with experimental data. In the present study we have used three approaches viz. HB, TG and BS to study phonon dynamics of Ca70Mg30 metallic glass. The local field correction function due to Sarkar et al is used for the first time to introduce the exchange and correlation effects in the aforesaid properties [13].

2. THEORETICAL METHODOLOGY The model potential used to explain the electron-ion interaction in the study of metallic glass is given by [13-15], ⎛ − Z e2 ⎡ r ⎞⎤ 2 ⎢ 2 − exp ⎜⎜1 − ⎟⎟ ⎥ r ; r ≤ rc 3 rc ⎣⎢ . ⎝ rc ⎠ ⎥⎦ 2 −Ze = ; r ≥ rc r

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W (r ) =

(1)

Where rc is the parameter of the potential and Z is the valency. This form has feature of a Coulombic term outside the core and varying cancellation due to a repulsive and attractive contributions to the potential within the core [13-15]. The wave number representation of the model potential is [13-15] ⎡⎧ 12 6U 2 18U 2 6U 4 U2 + − + ⎢ ⎪− 1 + 2 + 3 2 2 1+U U 1+U 2 1+U 2 1+U 2 ⎢⎪ ⎢⎨ 4 2 24U ⎢ ⎪ + 24U − ⎢ ⎪⎩ 1 + U 2 4 1 + U 2 4 ⎢ + ⎢ ⎢ 3U 3 6U ⎢ ⎧ 6 − 12 + U + 3U − + ⎪ 2 3 2 2 2 ⎢ 1+U 1+U 1+U 2 1+U 2 − 4 π Z e 2 ⎢ ⎪U U (q ) = ⎨ Ω 0 q 2 ε (q ) ⎢ ⎪ 18U 3 6U 36U 3 6U 5 + − + ⎢ ⎪− 4 2 4 2 3 2 4 1+U 1+U 1+U 2 ⎢⎩ 1 + U ⎢ + ⎢ 2 ⎢ ⎧ U − 1 ⎫⎪ ⎢ 24 U 2 exp (1) ⎪⎨ 4 ⎬ ⎢ ⎪⎩ 1 + U 2 ⎪⎭ ⎢ ⎢ ⎢ ⎣

(

(

W Beff

) (

(

) (

(

) ( )

⎤ ⎥

) (

) ⎪⎪ cos (U )⎥

) (

) (

)

)

(

⎫

) (

) (

)

3

⎬ ⎪ ⎪ ⎭

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎫ ⎥ 3 ⎪ ⎥ ⎪ ⎬ sin (U ) ⎥ ⎥ ⎪ ⎥ ⎪ ⎥ ⎭ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎦

(2)

Computation of Phonon Dispersion Curves of Ca70Mg30 Metallic Glass

87

Where U = qrC , rC , Z, ΩO and ε(q) is the potential parameter, valency, atomic volume and modified Hartree dielectric function of the glass, respectively. The detailed information of this potential is given in our earlier papers [13-15]. The fundamental ingredient, which goes into the calculation of the phonon dynamics of simple metallic glasses, is the effective interatomic pair potential. The effective interatomic pair potential for the amorphous binary alloys can be found the binary system as a one component metallic fluid, i.e. the concept of effective atom [3, 4, 8]. In this concept a simple binary disordered system AX B1− X can be looked upon as an assembly of the effective atom (i.e. one component system). The effective interaction in the glass can be written as [3, 4, 8]

⎡ Sin(qr ) ⎤ 2 ( ) F q eff ⎢ ⎥ q dq . ∫ ⎣ qr ⎦

⎛ Z eff2 e 2 ⎞ Ω oeff ⎟+ Veff (r ) = ⎜ ⎜ r ⎟ π2 ⎠ ⎝

(3)

Here Z eff and Ωo eff are the effective valence and atomic volume of the one component

fluid, respectively. The energy wave number characteristics Feff (q ) appearing in the equation (3) is written as

Feff (q ) = eff

Where WB

(q )

− Ω o eff q 2

16 π

eff B

W

[ε (q )− 1] (q ) {1 + [ε (q ) − 1][1− f (q )]} . eff H

2

eff H

is the effective bare ion potential,

(4)

eff

ε Heff (q ) the modified Hartree

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dielectric function and f eff (q ) the local field correction function due to Sarkar et al [13] to introduce the exchange and correlation effects. In the present study the pair correlation function g (r ) is found using the expressions given in the reference [8]. The three theories for studying of phonons in amorphous solids as developed by HB [11], TG [12] and BS [7, 10] have been employed for studying the longitudinal and transverse phonon frequencies in Ca70Mg30 glass. According to the HB, the expressions for longitudinal phonon frequency and transverse phonon frequency are [11],

⎡

ωL2 (q ) = ωE2 ⎢1 − ⎣

sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ , − + qσ (qσ )2 (qσ )3 ⎥⎦

(5)

3 cos(qσ ) 3 sin (qσ ) ⎤ , + (qσ )2 (qσ )3 ⎥⎦

(6)

⎡

ωT2 (q ) = ωE2 ⎢1 − ⎣

⎛ 4πρeff with ω E2 = ⎜ ⎜ 3M eff ⎝

⎞∞ ⎟ g (r )Veff'' (r ) r 2 dr is the maximum frequency. ⎟∫ ⎠0

88

Aditya M. Vora

The expressions for longitudinal phonon frequency and transverse phonon frequency as per TG approach are [12],

⎛ 4πρeff ⎜ M ⎝ eff

ωL2 (q ) = ⎜

⎡⎧ ⎞∞ ⎫ ⎟ dr g (r ) ⎢⎨r Veff' (r ) ⎛⎜1 − sin(qr ) ⎞⎟⎬ + r 2Veff' ' (r ) ⎜ ⎟ ⎟∫ qr ⎠⎭ ⎝ ⎠0 ⎣⎢⎩ ,

{

⎛ 1 sin(qr ) 2 cos(qr ) 2 sin(qr ) ⎞⎤ ⎟⎥ − rVeff' (r ) ⎜⎜ − − + qr (qr )2 (qr )3 ⎟⎠⎦⎥ ⎝3

(7)

}

⎛ 4πρ eff ⎜ M ⎝ eff

ωT2 (q ) = ⎜

⎡⎧ ⎞∞ ⎫ ⎟ dr g (r ) ⎢ ⎨r Veff' (r )⎛⎜1 − sin (qr ) ⎞⎟ ⎬ + r 2Veff'' (r ) − ∫ ⎜ ⎟ ⎟ qr ⎠ ⎭ ⎢⎣ ⎩ ⎝ ⎠0 . ⎤ ⎛ 1 2 cos(qr ) 2 sin (qr ) ⎞ ⎟⎥ rVeff' (r ) ⎜⎜ + + (qr )2 (qr )3 ⎟⎠⎦⎥ ⎝3

{

(8)

}

Recently BS [10] approach was modified by Shukla and Campnaha [7]. They were introduced screening effects. Then, with the above assumptions and modification, the dispersion equations for an amorphous material can be written as [7, 10]

ρeff ω (q) = 2 L

2 NCeff q2

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ρeff ωT2 (q ) =

(β I0 +δ I2 ) +

2 ke kTF q2 ε (q) G(qrS )

2

2 q2 + kTF ε (q)

2 N Ceff ⎛ 1 ⎞ ⎜ β I 0 + δ ( I 0 − I 2 )⎟ . 2 2 q ⎝ ⎠

,

(9) (10)

The other details of used constants in this approach were already narrated in literature [7, 10]. Here M eff is the effective atomic mass, ρ eff the effective number density and N Ceff the effective coordination number of the glassy system, respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ω L ∝ q and ωT ∝ q , ∴ ω L =υ L q and ωT =υ T q . Where

(11)

υ L and υT are the longitudinal and transverse sound velocities, in the glass

respectively. For the three approaches the equations are : For HB approach the formulations for υ L and υ T are given by [11],

υ L (HB ) = ω E and

3σ 2 , 10

(12)

Computation of Phonon Dispersion Curves of Ca70Mg30 Metallic Glass

σ2

υ T (HB ) = ω E

10

.

89

(13)

In TG approach the expressions for υ L and υ T are written by [12],

⎡ ⎛ 4π ρ eff υ L (TG) = ⎢ ⎜⎜ ⎢⎣ ⎝ 30 M eff

⎞ ⎟ ⎟ ⎠

∞

⎡ ⎛ 4π ρeff υT (TG) = ⎢ ⎜⎜ ⎢⎣ ⎝ 30 Meff

⎞ ⎟ ⎟ ⎠

∞

12

⎤ dr g(r ) r 3 { r V ′′(r ) − 4V ′ (r )}⎥ , ⎥⎦

∫ 0

(14)

and

∫ 0

12

⎤ dr g(r) r { 3r V ′′(r) − 4V ′ (r)}⎥ ⎥⎦ 3

.

(15)

The formulations for υL and υ T in BS approach are as follows [7, 10], 12

⎡ N Ceff ⎛ 1 1 ⎞ ke ⎤ υ L (BS ) = ⎢ ⎜ β+ δ ⎟+ ⎥ , 5 ⎠ 3⎦ ⎣ ρa ⎝ 3

(16)

and

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⎡ N Ceff ⎛ 1 1 υ T (BS ) = ⎢ ⎜ β+ δ 15 ⎣ ρa ⎝ 3

12

⎞⎤ ⎟⎥ , ⎠⎦

(17)

In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υ L and υ T are computed. The isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature θD are found using the expressions [8],

With

4 ⎞ ⎛ BT = ρ M ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(18)

G = ρ M υT2 .

(19)

ρ M is the isotropic number density of the solid. ⎛ υT2 ⎞ ⎟ 2 ⎟ ⎝ υL ⎠

σ = 1 − 2⎜⎜

⎛υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ , ⎝ υL ⎠

(20)

90

Aditya M. Vora

Y = 2G (σ + 1) ,

(21)

1

⎡ 9 ρ eff ⎤ 3 ⎡ 1 2 ⎤ = ωD = θD = 2π ⎢ = ⎥ ⎢ 3 + 3⎥ kB kB ⎣ 4 π ⎦ ⎣υ L υT ⎦ here

(−13 ) .

(22)

ωD is the Debye frequency.

The low temperature specific heat CV can be calculated from the following expressions given by Kovalenko and Krasny [16],

CV =

ΩOeff =2 kBT2

d3q

∑ ∫ (2π)

λ=L,T

3

ωλ2 (q)

⎡ ⎛ =ωλ (q) ⎞ ⎤ ⎡ ⎛ =ω (q) ⎞⎤ ⎟⎟ −1⎥ ⎢1− exp⎜⎜− λ ⎟⎟⎥ ⎢exp⎜⎜ ⎢⎣ ⎝ kB T ⎠ ⎥⎦ ⎢⎣ ⎝ kB T ⎠⎥⎦

.

(23)

3. RESULTS AND DISCUSSION

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The input parameters and other related constants used in the present computations are Zeff = 2.00, ΩOeff = 3.646 x 10-23 cm3, rceff = 6.663 x 10-9 cm, Nceff = 12.00, Meff = 5.871 x 10-23 gm and ρM = 1.6091 gm/cm3. The presently computed effective pair potential of homovalent Ca70Mg30 glass is shown in Figure 1 alongwith the other such results [1, 4, 6]. It is seen that the inclusion of exchange and correlation effect changes the nature of the pair potential, significantly. The first zero for V (r = r0 ) occurs at r0 ≤ 4.5 au. The well width and the position of Vmin (r ) are also affected by the nature of the screening. The maximum depths in the pair potential is obtained and move towards the left as compared to the potentials of Hafner [1] and Saxena et al.[4] The results of Saxena et al [4] show significant oscillations and potential energy remains positive in the large r–region. Thus Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions in their studies. The PDC of Ca70Mg30 due to three approaches (HB, TG, BS) is shown in Figure 2. It is observed from the figure that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. Moreover, the present outcome of both the phonon modes due to HB and TG approaches are more enhanced than those of BS approach. The first plunge in the longitudinal branch falls at q ≈ 1.8Å-1 for BS, q ≈ 2.6Å-1 for TG and q ≈ 2.7Å-1 for HB approach. The first crossover position of -1

ωL and ωT in the HB, TG and BS approaches is observed, respectively, at

2.1Å , 1.9Å-1 and 1.2Å-1. The presently computed values are compared with experimental [9] and theoretical results of Hafner [1] and others [3-6]. It is apparent that position of first peak obtained with the help of HB and TG approaches are closer to each other but their magnitudes are differing from each other substantially. Compared to the results reported by Hafner [1], which overestimates the experimental results, the present theoretical results of

Computation of Phonon Dispersion Curves of Ca70Mg30 Metallic Glass

91

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BS approach are found quite satisfactory with the experimental data of Suck et al [9] and theoretical results of others [3-6].

Figure 1. Pair Potential.

The PDC 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. Actually, Neutron Inelastic Scattering (NIS) experiments on Mg70Zn30 glass, by Suck et al [9] have exposed vigorously low-lying short wavelength collective density excitation at wave vector transfer where the structure factor shows its main peak, which are called phonon-roton states [9]. The difference

92

Aditya M. Vora

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in the magnitude of the minimum around 2 k F seems to be due to the fact that the concept of roton has not been taken into account theoretically.

Figure 2. Phonon Dispersion Curves.

As shown in Figure 3, the mode of calculating phonon frequencies affects the anomalous

behaviour (i.e. deviation from the T law) of the specific heat (CV ) . The reason behind the 3

anomalous behaviour may be due to the low frequency modes modify the generalized vibrational density of states of the glass with that of the polycrystal. The results due to BS approach give stip initial rise for increase in the temperature at low T region while the other two approaches are generating similar trend of CV T → T . This may be due to the nearest 2

neighbour atoms interaction assumption in the BS approach. Furthermore, the thermodynamic and elastic properties estimated from the elastic part of the PDC are tabulated in Table 1. The comparison with other such results [3, 4] favours the present calculation and suggests that the present model potential is capable for explaining the

Computation of Phonon Dispersion Curves of Ca70Mg30 Metallic Glass

93

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thermodynamic and elastic properties of Ca70Mg30 glass. Among the three approaches, BS approach is producing lower values of the velocities. The Bulk modulus reported by others is lower then present results of three approaches.

Figure 3. The Vibrational Part of the Specific Heat CV.

Table 1. Thermodynamic and Elastic properties of Ca70Mg30 metallic glass Properties υ L x 105 cm/sec

υT

x 105 cm/sec 11

2

BT x 10 dyne/cm G x 1011 dyne/cm2

σ

Y x 1011 dyne/cm2

θD

K

HB 5.7833

TG 6.3999

BS 5.1779

Others [3, 4] 5.66, 4.67

3.3390

3.8498

1.6787

3.55, 2.34

2.9900

3.4110

3.7097

1.80, 2.49

1.7940 0.2499 4.4850

2.3850 0.2165 5.8025

0.4534 0.4413 1.3071

1.49, 0.94 0.17, 0.33 3.50, 2.51

332.97

382.46

171.64

–

94

Aditya M. Vora

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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[15] [16]

J. Hafner, Phys. Rev., B27, (1983), 678. J. Hafner and S. S. Jaswal, J. Phys. F: Met. Phys., 18, (1988), L1. N. S. Saxena, Meeta Rani, Arun Pratap, Prabhu Ram and M. P. Saksena, Phys. Rev., B32, (1988), 8093. N. S. Saxena, Arun Pratap, Deepika Bhandari and M. P. Saksena, Mater. Sci. Engg., A134 (1991) 927. P. C. Agarwal, K. A. Aziz and C. M. Kachhava, Acta Phys Hung,, 72, (1992), 183; Phys Stat Sol (b), 178, (1993), 303. P. C. Agarwal and C. M. Kachhava, Phys. Stat. Sol. (b), 179, (1993), 365; Ind. J. Pure & Appl. Phys., 31, (1993), 528. M. M. Shukla and J. R. Campanha, Acta Phys. Pol., A94, (1998), 655. B. Y. Thakore, P. N. Gajjar and A. R. Jani, Bull. Mater. Sci., 23, (2000), 5. J. –B. Suck, H. Rudin, H. –J. Güntherodt and H. Beck, J. Phys. C: Solid State Phys. 13, (1980), L1045. B. Bhatia and R. N. Singh, Phys Rev., B31, (1985), 4751. J. Hubbard and J. L. Beeby, J. Phys. C: Solid State Phys., 2, (1969), 556. S. Takeno and M. Goda, Prog Thero Phys., 45, (1971), 331; Prog Thero Phys., 47, (1972), 790. P. N. Gajjar, A. M. Vora, A. R. Jani, Proceedings of the 9th Asia Pacific Physics Conference, Hanoi, Vietnam (October 25-31, 2004) (2006) 429. Aditya M. Vora, Chinese Phys. Lett., 23, (2006), 1872; J. Non-Cryst. Sol., 352, (2006), 3217; J. Mater. Sci., 42, (2007), 935. M. Vora, M. H. Patel, P. N. Gajjar, A. R. Jani, Solid State Phys., 46, (2003), 315. N. P. Kovalenko and Yu. P. Krasny, Physica B162, (1990), 115.

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp. 95-114© 2011 Nova Science Publishers, Inc.

Chapter 7

STUDY OF VIBRATIONAL DYNAMICS OF FE40NI40B20 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, Gujarat, India

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ABSTRACT The vibrational dynamics of Fe40Ni40B20 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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 [116]. The Fe40Ni40B20 bulk metallic glass is the most important candidate of transition metal∗

E-mail address: [email protected]; Tel.: +91-2832-256424.

96

Aditya M. Vora

metalloid group. The PDC of this glass has been theoretically studied by Gupta et al. [11] by HB approach [12] using experimental structure factor. The experimental data of PDC of this glass is not available in the literature. Therefore, we have reported vibrational properties of this glass using pseudopotential theory for the first time with more advanced screening functions. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic

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properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio

σ

and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 22],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ Vs (r ) = ⎜ S F (q ) ⎢ ⎥ q dq . ⎜ r ⎟ π2 ∫ ⎣ qr ⎦ ⎠ ⎝

(2)

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

97

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [22], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2

16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [17]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [17],

ε H (q ) = 1 +

⎛ 1 − η2 ⎞ m e2 1+ η ⎟ ;η = q ⎜ + 1 ln ⎟ ⎜ 2η 2 2 2k F 1− η 2π k = η F

⎝

(5)

⎠

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

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and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [17] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [18] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [18],

f (q ) =

q 2 ⎡ 0.1534 ⎤ ⎥. ⎢1 + π k F2 ⎦ 4 k F2 ⎣

(7)

98

Aditya M. Vora

The Ichimaru-Utsumi (IU) local field correction function [19] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎤⎪ ⎡ ⎛q⎞ ⎛ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ . (8) f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛q⎞ ⎥⎦ ⎪ ⎛ q ⎞ ⎢⎣ ⎝ kF ⎠ ⎝ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜⎜ ⎟⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

On the basis of Ichimaru-Utsumi (IU) local field correction function [19] local field correction function, Farid et al. (F) [20] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ ⎛q⎞ ⎛q⎞ ⎛q⎞ k ⎝ kF ⎠ ⎪ . (9) f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ 4 ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩ Based on equations (8-9), Sarkar et al. (S) [21] have proposed a simple form of local field correction function, which is of the form

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⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [19-21]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

W (r ) =

− Z e2 rC3

− Z e2 = r

⎡ ⎛ r ⎞⎤ 2 ⎢2 − exp ⎜⎜1 − ⎟⎟⎥ r ; ⎢⎣ ⎝ rC ⎠⎥⎦ ;

r ≤ rC .

(11)

r ≥ rC

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is assumed that the potential within the core should not be zero nor constant but it should very

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

99

as a function of r . Thus, the model potential has the novel feature of representing varying cancellation of potential within the core over and above its continuity at r = rC and weak nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows:

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ⎣ (Z ) ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 22]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ 10 ⎠ ⎝ N ⎠ ⎝

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

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transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from

100

Aditya M. Vora

the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1. Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass

Z

Fe40Ni40B20

3.40

ZS

Zd

1.50

6.00

Ω0 3

M (amu)

N

70.83

47.98

8.00

(au)

ρM (gm/cm3)

rC (au)

7.5884

0.5789

rd (au) 1.07

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 23],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and

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BS approaches are given below [12-16]. According to the Hubbard-Beeby (HB) [12], the expressions for longitudinal and transverse phonon frequencies are as follows,

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ ω2L (q ) = ω2E ⎢1 − − + qσ (qσ)2 (qσ)3 ⎥⎦ ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + (qσ)2 (qσ)3 ⎥⎦ ⎣

(21)

∞

⎛ 4πρ ⎞ 2 with ω = ⎜ ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0 2 E

Following to Takeno-Goda (TG) [13, 14], the wave vector (q ) dependent longitudinal

and transverse phonon frequencies are written as

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass ∞

⎛ 4πρ ⎞ ω (q) = ⎜ ⎟ ∫ dr g ( r ) ⎝ M ⎠0 2 L

⎡ ⎧⎪ ⎛ sin ( qr ) ⎞ ⎫⎪ 2 ⎢ ⎨r V ′ ( r ) ⎜ 1 − ⎟ ⎬ + { r V ′′ ( r ) − rV ′ ( r )} qr ⎠ ⎪⎭ ⎝ ⎣⎢ ⎪⎩ ⎛ 1 sin ( qr ) 2 cos ( qr ) 2sin ( qr ) ⎞ ⎤ − + ⎜ − ⎟⎥ 2 3 ⎜3 ⎟⎥ qr qr qr ( ) ( ) ⎝ ⎠⎦

101

,

(22)

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − qr ⎝ M ⎠0 ⎠⎭ ⎝ ⎣⎢ ⎩

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . (23) ⎟⎥ rV ′(r )} ⎜⎜ + + (qr )2 (qr )3 ⎟⎠⎥⎦ ⎝3

According to modified Bhatia-Singh (BS) [15, 16] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18],

k k 2 q 2ε (q ) G (qrS ) 2N ω (q ) = C2 (β I 0 + δ I 2 )+ e TF 2 2 ρq q + kTF ε (q ) 2 L

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ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [15, 16]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in the earlier papers [1-16]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature

θ D are found using the expressions [1-10],

102

Aditya M. Vora

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

θD

⎛ υ2 ⎞ 1 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = 2π ⎢ ⎥ = = kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (31)

here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that

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of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [24]

CV =

Ω0 = 2 kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎤ ⎛ ⎞ ( ) = ω q λ λ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎟⎟⎥ ⎢exp⎜⎜ k T k B T ⎠⎦ ⎠ ⎦⎣ ⎝ ⎣ ⎝ B

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. The computed interatomic pair potentials V (r ) are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

103

nature of the interatomic pair potential, significantly. The first zero of the interatomic pair potential V (r = r0 ) due to all local field correction functions occurs at r0 = 2.9 au, which is

very close to the rP value at which the pair correlation function g (r ) shows its first peak

[11]. The interatomic pair potential well width and its minimum position Vmin (r ) are also

affected by the nature of the screening. The maximum depth in the interatomic pair potential is obtained for F-function, while minimum is for H-screening function. The present results do not show oscillatory behaviour and potential energy remains negative in the large r - region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions, which show the waving shape of the interatomic pair potential after r = 8.7 au. Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Fe40Ni40B20

0.52

0.42

0.32

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V(r) (Ryd.)

0.22

0.12

0.02 0

5

10

15

20

-0.08

-0.18

-0.28 r (au)

Figure 1. Dependence on screening on pair potentials of Fe40Ni40B20 bulk metallic glass.

104

Aditya M. Vora

1.5

Fe40Ni40B20

1

g(r)

0.5

0

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0

5

10

15

20

-0.5 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-1

r (au) Figure 2. Dependence on screening on pair correlation function of Fe40Ni40B20 bulk metallic glass.

Hence, the interatomic pair potentials converge towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the s- and delectrons. Bretonnet and Derouiche [25] are observed that the repulsive part of V (r ) is drawn

lower and its attractive part is deeper due to the d-electron effect and the V (r ) is shifted

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

105

towards the lower r -values. Therefore, the present results are supported the d-electron effect as noted by Bretonnet and Derouiche [25]. The computed pair correlation function (PCF) g (r ) of this glass is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.29, 1.53, 1.50, 1.50 and 1.57 for H-, T-, IU-, F- and S-function, respectively. While the ratio (r3 r1 ) of the

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position of the third peak (r3) to that of the first peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 1.77, 2.11, 2.08, 2.08 and 2.15 for H-, T-, IU-, F- and S-function, respectively. The (r2 r1 ) ratio is close to the c/a ratio in close-packed hexagonal structure i.e. c/a = 1.63, which suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 7.9. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after this point. Actually, this long range order is normal and it may be due to the waving shape of the interatomic pair potential. The screening dependence of the phonon eigen frequencies for longitudinal and transverse phonon modes calculated using three approaches are shown in Figures 3-5. It is observed that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present results of the PDC due to H-, T- and F-function are lying between those due to IU- and S-screening in HB approach. While, the computed outcomes of the PDC due to T-, IU- and F-function are lying between those due to H- and S-screening in TG approach. Also in BS approach, the present results of the PDC due to T-, IU- and F-function are lying between those due to Hand S-screening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.6Å-1 for H-, T-, IU-, F- and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.8Å-1 for H-, q ≈ 2.5Å-1 for T-,

q ≈ 2.7Å-1 for IU- as well as F-function and q ≈ 2.4Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.4Å-1 for H-, T-, IU-, Fand S-function. The first maximum in the longitudinal branch

ωL

of HB approach is found

around at q ≈ 1.8Å for H-, T-, IU-, F- and S-function. While, the first maximum in the -1

longitudinal branch of TG approach is found around at q ≈ 1.3Å-1 for H-, T-, IU- as well as F-function and q ≈ 0.7Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.7Å-1 for most of the local field correction functions.

106

Aditya M. Vora 4.5

HB

Fe40Ni40B20

4

13

-1

ωL and ωT (in 10 Sec )

3.5

3

2.5

2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1.5

1

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0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 3. Dependence on screening on phonon dispersion curves of Fe40Ni40B20 bulk metallic glass using HB approach.

It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon modes due to HB as well as TG approaches are more enhanced than the BS approach. The first crossover position of 1

ωL

and

ωT in the HB and BS approaches is observed at q ≈ 2.8Å-

and q ≈ 1.4Å-1 for most of the local field correction functions, respectively. While, the first

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

ωL

107

ωT in the TG approach is observed at q ≈ 2.2Å-1 for H-, T-, IU- as well as F-function and q ≈ 1.8Å-1 for S-function. Here in transverse branch, the crossover position of

and

frequencies increase with the wave vector q and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [26] in which, it 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. 3.5

TG

Fe40Ni40B20

3

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13

-1

ωL and ωT (in 10 Sec )

2.5

2

1.5

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Fe40Ni40B20 bulk metallic glass using TG approach.

108

Aditya M. Vora

BS

9

Fe40Ni40B20

8

6

13

-1

ωL and ωT (in 10 Sec )

7

5

4

3

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2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Fe40Ni40B20 bulk metallic glass using BS approach.

As shown in Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) . The reason behind the anomalous behaviour may be due to the low frequency modes modify the generalized vibrational density of states of the glass with that of the polycrystal.

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

109

Fe40Ni40B20

HB 160

140

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

2

CV/T (10 J/(mol-K ))

120

-2

100

80

60

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40

20

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Fe40Ni40B20 bulk metallic glass using HB approach.

These modes are mainly responsible for the difference in the temperature dependence of the specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches. It is apparent from the nature that CV is more

110

Aditya M. Vora

sensitive to screening. The initial rise on the CV T values is observed for low temperature and then further increase of temperature give convergent value.

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

TG

2

CV/T (10 J/(mol-K ))

6

-2

4

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2

Fe40Ni40B20 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Fe40Ni40B20 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

111

0.18

Fe40Ni40B20

BS

0.15

-2

2

CV/T (10 J/(mol-K ))

0.12

0.09

0.06

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0.03

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Fe40Ni40B20 bulk metallic glass using BS approach.

Furthermore, the thermodynamic and elastic properties estimated from the elastic limit of the PDC are tabulated in Table 2. It is seen that the screening theory plays an important role in the prediction of the thermodynamic and elastic properties of this glass. As phonon modes obtained for BS approach is very high compared to HB and TG approaches, the thermodynamic and elastic properties obtained for BS approach is also higher in magnitude. The presently computed results of the thermodynamic and elastic properties are found to be in qualitative agreement with the available theoretical or experimental data [11] in the literature.

112

Aditya M. Vora Table 2. Thermodynamic and Elastic properties of Fe40Ni40B20 bulk metallic glass

υL x App.

10

5

10

5

BT x 1011

G x 1011

dyne/cm2

dyne/cm2

σ

Y x 1011 dyne/cm2

θD (K)

cm/s

cm/s

H

3.15

1.82

4.18

2.51

0.25

6.27

274.49

T

3.18

1.84

4.27

2.56

0.25

6.41

277.59

IU

3.48

2.01

5.12

3.07

0.25

7.67

303.76

F

3.47

2.00

5.08

3.05

0.25

7.62

302.61

S

0.40

0.23

0.07

0.04

0.25

0.10

35.25

H

4.78

2.29

12.04

3.97

0.35

10.73

349.90

T

4.53

2.54

9.03

4.91

0.27

12.46

384.99

IU

4.67

2.52

10.09

4.83

0.29

12.48

382.87

F

4.72

2.57

10.26

4.99

0.29

12.89

389.33

S

3.69

2.16

5.62

3.55

0.24

8.80

326.21

H

10.25

3.57

66.81

9.65

0.43

27.63

551.63

T

10.36

3.71

67.44

10.45

0.43

29.82

573.62

IU

10.33

3.68

67.23

10.28

0.43

29.35

568.93

F

10.33

3.69

67.25

10.32

0.43

29.45

570.00

S

10.37

3.73

67.55

10.54

0.43

30.04

575.83

Others [11]

3.96

2.34

0.61

0.393

0.233

1.132

353.45

Expt. [11]

4.47

-

1.773

-

0.341

1.60

350.00

HB

TG

BS

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SCR

υT x

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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While 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 is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties.

Study of Vibrational Dynamics of Fe40Ni40B20 Bulk Metallic Glass

113

From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study. The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

CONCLUSIONS

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Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. Vora Aditya M. Romanian J. Phys. 2008, Vol. , pp - (in press). Gupta A.; Bhandari D.; Jain K. C.; Saxena N. S. Phys. Stat. Sol. (b). 1996, Vol. 195, pp 367-374. Hubbard J.; Beeby J. L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352.

114 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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[24] [25] [26]

Aditya M. Vora Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. Bhatia A. B.; Singh R. N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. Shukla M. M.; Campanha J. R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999. Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621. Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373. Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp.115-135©2011 Nova Science Publishers, Inc.

Chapter 8

STUDY OF VIBRATIONAL DYNAMICS OF NI80B10SI20 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, Gujarat, India

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ABSTRACT The vibrational dynamics of Ni80B10Si20 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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 [116]. The Ni80B10Si20 metallic glass is the most important candidate of transition metal∗

E-mail address: [email protected]; Tel.: +91-2832-256424.

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Aditya M. Vora

metalloid group. In our literature survey we have not found any reports on the experimental as well as theoretical work based on pseudopotential theory related to PDC of this glass. But, the structural properties of this glass have been studied by Sulir and Pyka [11]. Therefore, the vibrational properties of this glass are reported for the first time. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ

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and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 22],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ Vs (r ) = ⎜ S F (q ) ⎢ ⎥ q dq . ⎜ r ⎟ π2 ∫ qr ⎣ ⎦ ⎠ ⎝

(2)

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

117

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [22], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2

16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [17]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [17],

ε H (q ) = 1 +

⎛ 1 − η2 ⎞ m e2 1+ η ⎟ ;η = q ⎜ ln + 1 ⎟ ⎜ 2η 2 2 1− η 2k F 2π k = η F

⎝

(5)

⎠

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

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and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [17] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [18] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [18],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ ⎢1 + ⎥. π k F2 ⎦ ⎣

(7)

The Ichimaru-Utsumi (IU) local field correction function [19] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self

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Aditya M. Vora

consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎤⎪ ⎡ ⎛q⎞ ⎛ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ . f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛q⎞ ⎥⎦ ⎪ ⎛ q ⎞ ⎢⎣ ⎝ kF ⎠ ⎝ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜⎜ ⎟⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(8)

On the basis of Ichimaru-Utsumi (IU) local field correction function [19] local field correction function, Farid et al. (F) [20] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ 2 − ⎜⎜ ⎟⎟ ⎪ 4⎜ ⎟ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

(9)

Based on equations (8-9), Sarkar et al. (S) [21] have proposed a simple form of local field correction function, which is of the form

⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎜ ⎝ kF ⎪⎩ ⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

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The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [19-21]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

⎛ r ⎞⎤ 2 − Z e2 ⎡ ⎜ W (r ) = 2 − exp 1 − ⎢ ⎜ r ⎟⎟⎥ r ; rC3 ⎣⎢ ⎥ C ⎠⎦ ⎝ =

−Z e r

2

;

r ≤ rC .

(11)

r ≥ rC

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is assumed that the potential within the core should not be zero nor constant but it should very as a function of r . Thus, the model potential has the novel feature of representing varying cancellation of potential within the core over and above its continuity at r = rC and weak

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

119

nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows :

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ⎣ (Z ) ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 22]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ ⎝ 10 ⎠ ⎝ N ⎠

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

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transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from

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Aditya M. Vora

the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1. Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass

Z

Ni80B10Si20

3.90

ZS

Zd

1.50

6.80

Ω0 3

M (amu)

N

77.77

50.85

9.60

(au)

ρM (gm/cm3)

rC (au)

7.3246

0.5450

rd (au) 1.54

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 23],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T ⎣⎝ B ⎠ ⎦

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and

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BS approaches are given below [12-16]. According to the Hubbard-Beeby (HB) [12], the expressions for longitudinal and transverse phonon frequencies are as follows,

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ ω2L (q ) = ω2E ⎢1 − − + 2 σ q ( ) (qσ)3 ⎥⎦ σ q ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + (qσ)2 (qσ)3 ⎥⎦ ⎣

(21)

∞

⎛ 4πρ ⎞ 2 with ω = ⎜ ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0 2 E

Following to Takeno-Goda (TG) [13, 14], the wave vector (q ) dependent longitudinal

and transverse phonon frequencies are written as

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

121

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎠⎭ ⎝ M ⎠0 ⎢⎣⎩ ⎝

{

ω L2 (q ) = ⎜

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ , (22) − + 2 3 ⎜3 ⎟ qr ( ) ( ) qr qr ⎝ ⎠⎥⎦

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − M qr ⎠0 ⎝ ⎠⎭ ⎝ ⎣⎢ ⎩

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . (23) ⎟⎥ + rV ′(r )} ⎜⎜ + 2 3 ⎟ 3 ( ) ( ) qr qr ⎝ ⎠ ⎦⎥

According to modified Bhatia-Singh (BS) [15, 16] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18],

k k 2 q 2ε (q ) G (qrS ) 2N ω (q ) = C2 (β I 0 + δ I 2 )+ e TF 2 2 ρq q + kTF ε (q ) 2 L

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ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [15, 16]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in the earlier papers [1-16]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT ,

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Aditya M. Vora

modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature

θ D are found using the expressions [1-10],

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

θD

⎛ υT2 ⎞ 1 − 2⎜⎜ 2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υT2 ⎞ 2 − 2⎜⎜ 2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = = = 2π ⎢ ⎥ kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (31)

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here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [24]

Ω0 = 2 CV = kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎛ = ω λ (q ) ⎞⎤ λ ⎟⎟⎥ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎢exp⎜⎜ k T k T B ⎠⎦ ⎝ ⎠ ⎦⎣ ⎣ ⎝ B

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

123

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. The computed interatomic pair potentials V (r ) are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the nature of the interatomic pair potential, significantly. The first zero of the interatomic pair potential V (r = r0 ) due to all local field correction functions occurs at r0 = 2.5 au, which is very close to the rP value at which the pair correlation function g (r ) shows its first peak. The interatomic pair potential well width and its minimum position Vmin (r ) are also affected

by the nature of the screening. The maximum depth in the interatomic pair potential is obtained for F-function, while minimum is for H-screening function. The present results do not show oscillatory behaviour and potential energy remains negative in the large r - region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions, which show the waving shape of the interatomic pair potential after r = 8.0 au. Hence, the interatomic pair potentials converge towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the s- and delectrons. Bretonnet and Derouiche [25] are observed that the repulsive part of V (r ) is drawn

lower and its attractive part is deeper due to the d-electron effect and the V (r ) is shifted

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towards the lower r -values. Therefore, the present results are supported the d-electron effect as noted by Bretonnet and Derouiche [25]. The pair correlation function (PCF) g (r ) computed theoretically through the interatomic pair potential is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.29, 1.54, 1.51, 1.51 and 1.54 for H-, T-, IU-, F- and S-function, respectively. While the ratio (r3 r1 ) of the position of the third peak (r3) to that of the first peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 1.78, 2.13, 2.09, 2.09 and 2.13 for H-, T-, IU-, F- and S-function, respectively. The (r2 r1 ) ratio is close 1.63 characteristic for the disordered bcc type crystallographic structures and therefore this component of the second peak might eventually be due to not complete amorphousation of the samples [11]. This suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 7.8 au. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after this point. In fact, this long range order is normal and it may be due to the waving shape of the interatomic pair potential.

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Aditya M. Vora

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Ni80B10Si10

0.52

0.42

V(r) (Ryd.)

0.32

0.22

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0.12

0.02 0

5

10

15

-0.08

-0.18

r (au) Figure 1. Dependence on screening on pair potentials of Ni80B10Si20 bulk metallic glass.

20

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

125

Ni80B10Si10 1.5

g(r)

1

0.5

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0 0

5

10

-0.5

15

20

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-1

r (au) Figure 2. Dependence on screening on pair correlation function of Ni80B10Si20 bulk metallic glass.

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Aditya M. Vora

The screening dependence of the phonon eigen frequencies for longitudinal and transverse phonon modes calculated using three approaches are shown in Figures 3-5. It is observed that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present results of the PDC due to H-, T- and F-function are lying between those due to IU- and S-screening in HB approach. While, the computed outcomes of the PDC due to T-, IU- and F-function are lying between those due to H- and S-screening in TG approach. Also in BS approach, the present results of the PDC due to T-, IU- and F-function are lying between those due to Hand S-screening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.5Å-1 for H-, T-, IU-, F- and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.7Å-1 for H-, q ≈ 2.3Å-1 for T, q ≈ 2.5Å-1 for IU- as well as F-function and q ≈ 1.9Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.4Å-1 for H-, T-, IU-, F- and S-function. The first maximum in the longitudinal branch

ωL

of HB

approach is found around at q ≈ 1.8Å for H-, T-, IU-, F- and S-function. While, the first -1

maximum in the longitudinal branch of TG approach is found around at q ≈ 1.2Å-1 for H-, T-, IU- as well as F-function and q ≈ 0.7Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.6Å-1 for most of the local

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field correction functions. It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon modes due to HB as well as TG approaches are more enhanced than the BS approach. The first crossover position of BS approaches is observed at q ≈ 2.7Å

-1

and q ≈ 1.4Å

-1

ωL

and

ωT in the HB and

for most of the local field

correction functions, respectively. While, the first crossover position of

ωL

and

ωT in the

TG approach is observed at q ≈ 2.1Å for H-, q ≈ 1.7 for T-, q ≈ 1.9 for IU- as well as F-1

function and q ≈ 1.4Å-1 for S-function. Here in transverse branch, the frequencies increase with the wave vector q and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [26] in which, it 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 Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) .

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

127

3

Ni80B10Si10

HB

2

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13

-1

ωL and ωT (in 10 Sec )

2.5

1.5

1

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 3. Dependence on screening on phonon dispersion curves of Ni80B10Si20 bulk metallic glass using HB approach.

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Aditya M. Vora

Ni80B10Si10

TG 2

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13

-1

ωL and ωT (in 10 Sec )

1.5

1

0.5 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

1

2

0-1

q (A

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Ni80B10Si20 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

129

9

BS

Ni80B10Si10

8

13

-1

ωL and ωT (in 10 Sec )

7

6

5

4

3

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2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Ni80B10Si20 bulk metallic glass using BS approach.

The reason behind the anomalous behaviour may be due to 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 specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches. It is apparent from the nature that CV is more

130

Aditya M. Vora

sensitive to screening. The initial rise on the CV T values is observed for low temperature and then further increase of temperature give convergent value.

20 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

HB

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-2

2

CV/T (10 J/(mol-K ))

15

10

5

Ni80B10Si10 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Ni80B10Si20 bulk metallic glass using HB approach.

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

131

4

TG

Ni80B10Si10

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-2

2

CV/T (10 J/(mol-K ))

3

2

1 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Ni80B10Si20 bulk metallic glass using TG approach.

132

Aditya M. Vora

BS

Ni80B10Si10 0.12

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-2

2

CV/T (10 J/(mol-K ))

0.09

0.06

0.03 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Ni80B10Si20 bulk metallic glass using BS approach.

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

133

The thermodynamic and elastic properties estimated from the elastic limit of the PDC are narrated in Table 2. It is seen that the results due to HB and TG approaches are very low in comparison with BS approach. The effect of various local field correction functions is also distinguishable on the thermodynamic and elastic properties of Ni80B10Si20 glass. Table 2. Thermodynamic and Elastic properties of Ni80B10Si20bulk metallic glass

υL x App.

HB

TG

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BS

SCR H T IU F S H T IU F S H T IU F S

5

10 cm/s 2.15 2.48 2.61 2.62 1.38 4.26 4.68 4.67 4.75 3.97 10.06 10.19 10.17 10.18 10.19

υT x 5

10 cm/s 1.24 1.43 1.51 1.51 0.79 2.32 2.71 2.67 2.72 2.32 2.49 2.71 2.68 2.70 2.71

BT x 1011

G x 1011

dyne/cm2

dyne/cm2

1.88 2.51 2.77 2.79 0.77 8.05 8.88 9.02 9.29 6.31 68.12 68.91 68.74 68.83 68.87

1.13 1.51 1.66 1.67 0.46 3.93 5.39 5.22 5.41 3.93 4.54 5.39 5.28 5.35 5.37

σ 0.25 0.25 0.25 0.25 0.25 0.29 0.25 0.26 0.26 0.24 0.47 0.46 0.46 0.46 0.46

Y x 1011 dyne/cm2

θD (K)

2.82 3.77 4.15 4.18 1.15 10.14 13.45 13.12 13.60 9.77 13.32 15.77 15.44 15.66 15.69

181.81 209.96 220.48 221.14 116.25 340.80 397.07 391.06 398.24 338.98 374.77 408.33 404.00 406.88 407.31

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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While 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 is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties. From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study.

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Aditya M. Vora

The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. Vora Aditya M. Romanian J. Phys. 2008, Vol. , pp - (in press). Sulir L. D.; Pyka M. Acta Phys. Pol. A. 1983, Vol. 63, pp 747-754. Hubbard J.; Beeby J. L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352. Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. Bhatia A. B.; Singh R. N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. Shukla M. M.; Campanha J. R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999.

Study of Vibrational Dynamics of Ni80B10Si20 Bulk Metallic Glass

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[18] Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. [19] Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. [20] Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621. [21] Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. [22] Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373. [23] Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. [24] Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. [25] Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. [26] Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

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In: Bulk Metallic Glasses ISBN: 978-1-61122-938-7 Editors: T. George, R. Letfullin and G. Zhang, pp.137-156©2011 Nova Science Publishers, Inc.

Chapter 9

STUDY OF VIBRATIONAL DYNAMICS OF FE60NI20B10SI10 BULK METALLIC GLASS Aditya M. Vora∗ Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch, 370 001, Gujarat, India

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ABSTRACT The vibrational dynamics of Fe60Ni20B10Si10 bulk metallic glass has been studied at room temperature in terms of phonon eigen frequencies of longitudinal and transverse modes employing three different approaches proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS). The well recognized model potential is employed successfully to explain electron-ion interaction in the metallic glass. Instead of using experimental values of the pair correlation function g(r), which is generated from the computed pair potential. The present findings of phonon dispersion curve are found in fair agreement with available theoretical as well as experimental data. The thermodynamic properties obtained by HB and TG approaches are found very lower than those obtained by BS approach. The pseudo-alloyatom (PAA) model is applied for the first time instead of Vegard's Law.

Keywords: Pair potential, Bulk Metallic Glasses, Phonon dispersion curves, Thermal properties, Elastic properties PACS: 63.50. +x, 65.60. +a

1. INTRODUCTION The mankind has been manufacturing glassy materials for several thousands 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 [116]. The Fe60Ni20B10Si10 metallic glass is the most important candidate of transition metal∗

E-mail address: [email protected]; Tel.: +91-2832-256424.

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Aditya M. Vora

metalloid group. In our literature survey we have not found any reports on the experimental as well as theoretical work based on pseudopotential theory related to PDC of this glass. But, the structural properties of this glass have been studied by Sulir and Pyka [11]. Therefore, the vibrational properties of this glass are reported for the first time. In most of the theoretical studies, the pseudopotential parameter is evaluated such that it generates a pair correlation function, which is in good agreement with experimental data available in the literature. In most of these studies, the Vegard's law was used to explain electron-ion interaction for binaries. But it is well known that PAA is a more meaningful approach to explain such kind of interactions in metallic alloys and metallic glasses [1-10]. Hence, in the present article the PAA model is used to investigate the vibrational dynamics of fA + gB + hC + iD bulk glassy system. This article introduces pseudopotential based theory to address the problems of vibrational dynamics of bulk metallic glasses and their related elastic as well as thermodynamic properties with the method of computing the properties under investigation. Three main theoretical approaches given by Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] are used in the present study for computing the phonon frequencies of the bulk non-crystalline or glassy alloys. Five local field correction functions viz. Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are used for the first time in the present investigation to study the screening influence on the aforesaid properties of bulk metallic glasses. Besides, the thermodynamic properties such as longitudinal sound velocity υ L , transverse sound velocity

υ T and Debye temperature θ D , low temperature specific heat capacity CV and some elastic properties viz. the isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ

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and Young’s modulus Y are also calculated from the elastic part of the phonon dispersion curves (PDC).

2. THEORETICAL METHODOLOGY The fundamental ingredient, which goes into the calculation of the vibrational dynamics of bulk metallic glasses, is the pair potential. In the present study, for bulk metallic glasses, the pair potential is computed using [1-10, 22],

V (r ) = VS (r ) + Vb (r ) + Vr (r ) .

(1)

The s-electron contribution to the pair potential VS (r ) is calculated from [1-10],

⎛ Z 2 e 2 ⎞ ΩO ⎡ Sin(qr ) ⎤ 2 ⎟+ Vs (r ) = ⎜ S F (q ) ⎢ ⎥ q dq . ⎜ r ⎟ π2 ∫ ⎣ qr ⎦ ⎠ ⎝

(2)

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

139

Here Z S ~1.5 is found by integrating the partial s-density of states resulting from selfconsistent band structure calculation for the entire 3d and 4d series [22], while ΩO is the effective atomic volume of the one component fluid. The energy wave number characteristics appearing in the equation (2) is written as [1-10]

F (q ) =

− ΩO q 2

16 π

WB (q )

Here, WB (q ) is the bare ion potential

2

[ε H (q ) − 1] . {1 + [ε H (q ) − 1][1− f (q )]}

(3)

ε H (q ) the modified Hartree dielectric function,

which is written as [17]

ε (q ) = 1 + (ε H (q ) − 1) (1 − f (q )) .

(4)

While, ε H (X ) is the static Hartree dielectric function and the expression of it is given by [17],

ε H (q ) = 1 +

m e2 2 π k = 2 η2 F

⎛ 1 − η2 ⎞ 1+ η ⎟ ;η = q ⎜ ln + 1 ⎟ ⎜ 2η 1− η 2k F ⎝ ⎠

(5)

here m, e, = are the electronic mass, the electronic charge, the Plank’s constant, respectively

(

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and k F = 3π Z Ω O 2

)

12

is the Fermi wave vector, in which Z the valence. While f (q ) is

the local field correction function. In the present investigation, the local field correction functions due to Hartree (H) [17], Taylor (T) [18], Ichimaru-Utsumi (IU) [19], Farid et al. (F) [20] and Sarkar et al. (S) [21] are incorporated to see the impact of exchange and correlation effects on the aforesaid properties. The details of all the local field corrections are below. The Hartree (H) screening function [17] is purely static, and it does not include the exchange and correlation effects. The expression of it is,

f (q ) = 0 .

(6)

Taylor (T) [18] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor (T) [18],

f (q ) =

q2 4 k F2

⎡ 0.1534 ⎤ ⎢1 + ⎥. π k F2 ⎦ ⎣

(7)

The Ichimaru-Utsumi (IU) local field correction function [19] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self

140

Aditya M. Vora

consistency condition in the compressibility sum rule and short range correlations. The fitting formula is ⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎛ ⎤⎪ k ⎛q⎞ ⎛q⎞ 8 A ⎞⎛ q ⎞ ⎝ kF ⎠ ⎪ .(8) f (q) = AIU ⎜⎜ ⎟⎟ + BIU ⎜⎜ ⎟⎟ + CIU + ⎢ AIU ⎜⎜ ⎟⎟ + ⎜ BIU + IU ⎟ ⎜⎜ ⎟⎟ − CIU ⎥ ⎨ ⎝ F ⎠ ln ⎬ 3 ⎠ ⎝ kF ⎠ ⎛q⎞ ⎢⎣ ⎝ kF ⎠ ⎝ ⎥⎦ ⎪ ⎛ q ⎞ ⎝ kF ⎠ ⎝ kF ⎠ 4⎜⎜ ⎟⎟ 2 − ⎜⎜ ⎟⎟ ⎪ ⎪ ⎝ kF ⎠ ⎝ kF ⎠ ⎪⎭ ⎩

On the basis of Ichimaru-Utsumi (IU) local field correction function [19] local field correction function, Farid et al. (F) [20] have given a local field correction function of the form

⎧ ⎛ q ⎞2 ⎛ q ⎞⎫ ⎪ 4 − ⎜⎜ ⎟⎟ 2 + ⎜⎜ ⎟⎟ ⎪ 4 2 4 2 ⎡ ⎛q⎞ ⎤⎪ k ⎛q⎞ ⎛q⎞ ⎛q⎞ ⎝ kF ⎠ ⎪ . (9) f (q) = AF ⎜⎜ ⎟⎟ + BF ⎜⎜ ⎟⎟ + CF + ⎢ AF ⎜⎜ ⎟⎟ + DF ⎜⎜ ⎟⎟ − CF ⎥ ⎨ ⎝ F ⎠ ln ⎬ ⎛ q ⎞⎪ ⎢⎣ ⎝ kF ⎠ ⎥⎦ ⎪ ⎛⎜ q ⎞⎟ ⎝ kF ⎠ ⎝ kF ⎠ ⎝ kF ⎠ ⎟ ⎜ 2−⎜ ⎟ 4 ⎪ ⎜⎝ kF ⎟⎠ ⎝ kF ⎠ ⎪⎭ ⎩ Based on equations (8-9), Sarkar et al. (S) [21] have proposed a simple form of local field correction function, which is of the form

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⎧⎪ ⎛ ⎛ q f (q ) = AS ⎨1 − ⎜1 + BS ⎜⎜ ⎝ kF ⎪⎩ ⎜⎝

⎞ ⎟⎟ ⎠

4

⎞ ⎛ ⎟ exp ⎜ − C ⎛⎜ q S⎜ ⎟ ⎜ ⎝ kF ⎠ ⎝

⎞ ⎟⎟ ⎠

2

⎞⎫⎪ ⎟ . ⎟⎬⎪ ⎠⎭

(10)

The parameters AIU , B IU , C IU , AF , B F , C F , D F , AS , BS and C S are the atomic volume dependent parameters of IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [19-21]. The well recognized model potential W B (r ) [1-10] (in r -space) used in the present computation is of the form,

W (r ) = =

⎛ r ⎞⎤ 2 − Z e2 ⎡ ⎢2 − exp ⎜⎜1 − ⎟⎟⎥ r ; 3 rC ⎣⎢ ⎝ rC ⎠⎦⎥ −Z e r

2

;

r ≤ rC .

(11)

r ≥ rC

This form has feature of a Coulombic term out side the core and varying cancellation due to a repulsive and an attractive contribution to the potential within the core. Hence it is assumed that the potential within the core should not be zero nor constant but it should very as a function of r . Thus, the model potential has the novel feature of representing varying

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

141

cancellation of potential within the core over and above its continuity at r = rC and weak nature [1-10]. Here rC is the parameter of the model potential of bulk metallic glasses. The model potential parameter rC is calculated from the well known formula [1-10] as follows:

⎡ 0.51 rS ⎤ . rC = ⎢ 13 ⎥ ⎣ (Z ) ⎦

(12)

Here rS is the Wigner-Seitz radius of the bulk metallic glasses. The d-electron contributions to the pair potential are expressed in terms of the number of d-electron Z d , the d-state radii rd and the nearest-neighbour coordination number N as follows [1-10, 22]:

⎛ Z ⎞ ⎛ 12 ⎞ Vb (r ) = − Z d ⎜1 − d ⎟ ⎜ ⎟ ⎝ 10 ⎠ ⎝ N ⎠

1

2

3

⎛ 28.06 ⎞ 2 rd ⎜ ⎟ 5 , ⎝ π ⎠ r

(13)

and 6

⎛ 450 ⎞ r Vr (r ) = Z d ⎜ 2 ⎟ d8 . ⎝π ⎠r

(14)

Here, Vb (r ) takes into account the Friedel-model band broadening contribution to the

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transition metal cohesion and Vr (r ) arises from the repulsion of the d-electron muffin-tin

orbital on different sites due to their non-orthogonality. Wills and Harrison (WH) [22] have studied the effect of the s- and d-bands. The parameters Z d , Z S , rd and N can be calculated by the following expressions,

Z d = fZ dA + g Z dB + hZ dC + iZ dD ,

(15)

Z S = fZ SA + g Z SB + hZ SC + iZ SD ,

(16)

rd = frdA + g rdB + hrdC + irdD ,

(17)

N = fN A + g N B + hN C + iN D ,

(18)

and

Where A , B , C and D are denoted the first, second, third and forth pure metallic components of the bulk metallic glasses while f , g , h and i the concentration factor of the first, second, third and forth metallic components. Z d , Z S and rd are determined from

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Aditya M. Vora

the band structure data of the pure component available in the literature [24]. The values used in the present study are listed in Table 1. Table 1. Input parameters and constants used in the present computation Bulk Metallic Glass

Z

Fe60Ni20B10Si10

4.30

ZS

Zd

1.65

6.80

Ω0 3

M (amu)

N

91.19

53.66

9.60

(au)

ρM (gm/cm3)

rC (au)

6.5919

0.5385

rd (au) 1.07

A quantity which is equally important as the pair potential while studying a disorder system is the pair correlation function g (r ) . In the present study the pair correlation function g(r) can be computed from the relation [1-10, 23],

⎡⎛ − V (r ) ⎞ ⎤ ⎟⎟ − 1⎥ . g (r ) = exp⎢⎜⎜ k T B ⎝ ⎠ ⎦ ⎣

(19)

Here k B is the Boltzmann’s constant and T the room temperature of the amorphous system. The theories of Hubbard-Beeby (HB) [12], Takeno-Goda (TG) [13, 14] and Bhatia-Singh (BS) [15, 16] have been employed in the present computation. The expressions for longitudinal phonon frequency ω L and transverse phonon frequency ω T as per HB, TG and

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BS approaches are given below [12-16]. According to the Hubbard-Beeby (HB) [12], the expressions for longitudinal and transverse phonon frequencies are as follows,

⎡ sin (qσ ) 6 cos(qσ ) 6 sin (qσ ) ⎤ ω2L (q ) = ω2E ⎢1 − − + qσ (qσ)2 (qσ)3 ⎥⎦ ⎣

(20)

⎡ 3 cos(qσ ) 3 sin (qσ ) ⎤ ωT2 (q ) = ω2E ⎢1 − + (qσ)2 (qσ)3 ⎥⎦ ⎣

(21)

∞

⎛ 4πρ ⎞ 2 with ω = ⎜ ⎟ ∫ g (r )V ′′(r ) r dr is the maximum frequency. ⎝ 3M ⎠ 0 2 E

Following to Takeno-Goda (TG) [13, 14], the wave vector (q ) dependent longitudinal

and transverse phonon frequencies are written as

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

143

∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎟⎬ + r 2V ′′(r ) − rV ′(r ) ⎟ ∫ dr g (r ) ⎢⎨r V ′(r ) ⎜⎜1 − qr ⎠⎭ ⎝ M ⎠0 ⎢⎣⎩ ⎝

{

ω L2 (q ) = ⎜

}

⎛ 1 sin (qr ) 2 cos(qr ) 2 sin (qr ) ⎞⎤ ⎜ − ⎟⎥ , (22) − + 2 3 ⎜3 ⎟ qr ( ) ( ) qr qr ⎝ ⎠⎥⎦ ∞ ⎡⎧ ⎛ sin (qr ) ⎞⎫ ⎛ 4πρ ⎞ ⎟⎬ + r 2V ′′(r ) − ⎟ ∫ dr g (r ) ⎢ ⎨r V ′(r ) ⎜⎜1 − qr ⎟⎠⎭ ⎝ M ⎠0 ⎢⎣ ⎩ ⎝

{

ω T2 (q ) = ⎜

⎛ 1 2 cos (qr ) 2 sin (qr ) ⎞ ⎤ . (23) ⎟⎥ rV ′(r )} ⎜⎜ + + (qr )2 (qr )3 ⎟⎠⎥⎦ ⎝3

According to modified Bhatia-Singh (BS) [15, 16] approach, the phonon frequencies of longitudinal and transverse branches are given by Shukla and Campanha [18], 2 ke kTF q 2ε (q ) G (qrS ) 2 NC (β I 0 + δ I 2 ) + ω (q ) = 2 ρ q2 q 2 + kTF ε (q ) 2 L

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ωT2 (q ) =

2 NC ⎛ 1 ⎞ β I 0 + δ (I 0 − I 2 )⎟ 2 ⎜ 2 ρq ⎝ ⎠

2

(24)

(25)

Other details of the constants used in this approach were already narrated in literature [15, 16]. Here M , ρ are the atomic mass and the number density of the glassy component

while V′(r ) and V′′(r ) be the first and second derivative of the effective pair potential,

respectively. In the long-wavelength limit of the frequency spectrum, the both the frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relationships,

ωL ∝ q and ωT ∝ q ,

∴ ωL = υL q

and ωT = υT q .

(26)

Where υL and υT are the longitudinal and transverse sound velocities of the glassy alloys, respectively. The mathematical expressions of υL and υT are given in the earlier papers [1-16]. In the long-wavelength limit of the frequency spectrum, transverse and longitudinal sound velocities υL and υT are computed. The isothermal bulk modulus BT , modulus of rigidity G , Poisson’s ratio σ , Young’s modulus Y and the Debye temperature

θ D are found using the expressions [1-10],

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Aditya M. Vora

With

4 ⎞ ⎛ BT = ρ ⎜υ L2 − υT2 ⎟ , 3 ⎠ ⎝

(27)

G = ρ υT2 .

(28)

ρ is the isotropic number density of the solid.

θD

⎛ υ2 ⎞ 1 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠ , σ = ⎛ υ2 ⎞ 2 − 2⎜⎜ T2 ⎟⎟ ⎝ υL ⎠

(29)

Y = 2G (σ + 1) ,

(30)

⎡9 ρ ⎤ = ωD = = = 2π ⎢ ⎥ kB kB ⎣ 4π ⎦

1

3

⎡1 2⎤ ⎢ 3 + 3⎥ ⎣υ L υ T ⎦

(−13 ) (31)

here ωD is the Debye frequency. Now a day it is firmly established that for all amorphous solids, the universal temperature behaviour of vibrational contributions to the heat capacity (CV ) differs essentially from that

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of crystal. In the thermodynamic limit ( N O → 0 , Ω 0 → 0 and N O Ω 0 = constant, with

N O is the number of atoms in the unit cell), one can obtain [24]

CV =

Ω0 = 2 kB T 2

ω λ2 (q ) d 3q . ∑λ ∫ (2π )3 ⎡ ⎛ = ω (q ) ⎞ ⎤ ⎡ ⎛ = ω λ (q ) ⎞⎤ λ ⎟⎥ ⎟⎟ −1⎥ ⎢1 − exp⎜⎜ − ⎢exp⎜⎜ k B T ⎟⎠⎦ ⎝ ⎣ ⎝ kB T ⎠ ⎦ ⎣

(32)

Where, T is the temperature of the system, respectively. The basic features of temperature dependence of CV are determined by the behavior of ω L (q ) and ωT (q ) .

3. RESULTS AND DISCUSSION The input parameters and other related constants used in the present computations are written in Table 1. The computed interatomic pair potentials V (r ) are shown in Figure 1. It is seen that the inclusion of exchange and correlation effects in the static H-dielectric screening changes the nature of the interatomic pair potential, significantly. The first zero of the interatomic pair

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

145

potential V (r = r0 ) due to all local field correction functions occurs at r0 = 2.6 au. The interatomic pair potential well width and its minimum position Vmin (r ) are also affected by

the nature of the screening. The maximum depth in the interatomic pair potential is obtained for F-function, while minimum is for H-screening function. The present results do not show oscillatory behaviour and potential energy remains negative in the large r - region. Thus, the Coulomb repulsive potential part dominates the oscillations due to ion-electron-ion interactions, which show the waving shape of the interatomic pair potential after r = 7.8 au.

Fe60Ni20B10Si20

0.53

0.43

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

V(r) (Ryd.)

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0.33

0.23

0.13

0.03 0

5

10

15

20

-0.07

-0.17

r (au) Figure 1. Dependence on screening on pair potentials of Fe60Ni20B10Si10 bulk metallic glass.

146

Aditya M. Vora

Hence, the interatomic pair potentials converge towards a finite value instead of zero in attractive region. All the interatomic pair potentials show the combined effect of the s- and delectrons. Bretonnet and Derouiche [25] are observed that the repulsive part of V (r ) is drawn

lower and its attractive part is deeper due to the d-electron effect and the V (r ) is shifted

towards the lower r -values. Therefore, the present results are supported the d-electron effect as noted by Bretonnet and Derouiche [25]. The pair correlation function (PCF) g (r ) computed theoretically through the interatomic pair potential is shown in Figure 2. It is found that, the peak positions due to S-function show higher while those due to H-function show lower. The screening effect is also observed in the nature of the PCF. The ratio (r2 r1 ) of the position of the second peak (r2) to that of the first peak (r1) is found 1.28, 1.53, 1.53, 1.53 and 1.57 for H-, T-, IU-, F- and S-function, respectively. While the ratio (r3 r1 ) of the position of the third peak (r3) to that of the first

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peak (r1) i.e. the ratio of the third atomic shell radius to the nearest-neighbour distance is found 1.75, 2.13, 2.13, 2.13 and 2.17 for H-, T-, IU-, F- and S-function, respectively. The (r2 r1 ) ratio is close 1.63 characteristic for the disordered bcc type crystallographic structures and therefore this component of the second peak might eventually be due to not complete amorphousation of the samples [11]. This suggests that the short range order of nearest neighbours is influenced more or less by the atomic arrangement of the crystalline structure. This result is typical of a metallic glass with a large main peak at the nearest-nearest distance followed by smaller peaks corresponding to more distant neighbours. The computed pair correlation function using various local field correction functions are overlapped with each other after r = 7.9 au. Therefore, it is very difficult to draw the remarks regarding the disorder is invisible after this point. In fact, this long range order is normal and it may be due to the waving shape of the interatomic pair potential. The screening dependence of the phonon eigen frequencies for longitudinal and transverse phonon modes calculated using three approaches are shown in Figures 3-5. It is observed that the inclusion of exchange and correlation effect enhances the phonon frequencies in both longitudinal as well as transverse branches. The present results of the PDC due to H-, T- and F-function are lying between those due to IU- and S-screening in HB approach. While, the computed outcomes of the PDC due to T-, IU- and F-function are lying between those due to H- and S-screening in TG approach. Also in BS approach, the present results of the PDC due to T-, IU- and F-function are lying between those due to H- and Sscreening. The first minimum in the longitudinal branch of HB approach is found around at q ≈ 3.3Å-1 for H-, T-, IU-, F- and S-function. While, the first minimum in the longitudinal branch of TG approach is found around at q ≈ 2.5Å-1 for H-, q ≈ 2.1Å-1 for T-, q ≈ 2.3Å-1 for IU- as well as F-function and q ≈ 1.9Å-1 for S-function. The first minimum in the longitudinal branch of BS approach is found around at q ≈ 1.4Å-1 for H-, T-, IU-, F- and S-function. The first maximum in the longitudinal branch

ωL

of HB approach is found around at q ≈ 1.7Å-1

for H-, T-, IU-, F- and S-function. While, the first maximum in the longitudinal branch of TG approach is found around at q ≈ 1.0Å-1 for H-, T-, IU- as well as F-function and q ≈ 0.8Å-1 for S-function. The first maximum in the longitudinal branch of BS approach is found around at q ≈ 0.6Å-1 for most of the local field correction functions.

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

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3

Fe60Ni20B10Si20 2.5

2

g(r)

1.5

1

0.5

0 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

0

5

10

15

20

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

-0.5

-1

r (au) Figure 2. Dependence on screening on pair correlation function of Fe60Ni20B10Si10 bulk metallic glass.

It is also observed from the Figures 3-5 that, the oscillations are more prominent in the longitudinal phonon modes as compared to the transverse modes in all three approaches. 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. Moreover, the present outcomes of both the phonon modes due to HB as well as TG approaches are more enhanced than the BS approach.

148

Aditya M. Vora

HB

3

Fe60Ni20B10Si20

-1

ωL and ωT (in 10 Sec )

2.5

13

2

1.5

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1 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0.5

0 0

1

2

q (A

0-1

3

4

5

)

Figure 3. Dependence on screening on phonon dispersion curves of Fe60Ni20B10Si10 bulk metallic glass using HB approach.

The first crossover position of

ωL

and

ωT in the HB and BS approaches is observed at

q ≈ 2.5Å and q ≈ 1.3Å for most of the local field correction functions, respectively. -1

-1

ωL

ωT in the TG approach is observed at q ≈ 2.0Å for H-, q ≈ 1.5 for T-, q ≈ 1.8 for IU- as well as F-function and q ≈ 1.5Å-1 for

While, the first crossover position of

and

-1

S-function. Here in transverse branch, the frequencies increase with the wave vector q

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

149

and then saturates at ≈ q = 2.0Å-1, which supports the well known Thorpe model [26] in which, it 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.

TG

Fe60Ni20B10Si20

-1

ωL and ωT (in 10 Sec )

2

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13

1.5

1

0.5

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

1

2

q (A

0-1

3

4

5

)

Figure 4. Dependence on screening on phonon dispersion curves of Fe60Ni20B10Si10 bulk metallic glass using TG approach.

150

Aditya M. Vora

BS

Fe60Ni20B10Si20

8

7

13

-1

ωL and ωT (in 10 Sec )

6

5

4

3

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2 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

1

0 0

1

2

q (A

0-1

3

4

5

)

Figure 5. Dependence on screening on phonon dispersion curves of Fe60Ni20B10Si10 bulk metallic glass using BS approach.

As shown in Figures 6-8, the exchange and correlation functions also affect the 3

anomalous behaviour (i.e. deviation from the T law) which is observed in the specific heat (CV ) . The reason behind the anomalous behaviour may be due to 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

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

151

specific heat which departs from the normal behaviour. At low temperature region high bump is observed in HB, TG and BS approaches. It is apparent from the nature that CV is more sensitive to screening. The initial rise on the CV T values is observed for low temperature and then further increase of temperature give convergent value.

15 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

HB

12.5

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-2

2

CV/T (10 J/(mol-K ))

10

7.5

5

2.5

Fe60Ni20B10Si20 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 6. Dependence on screening on low temperature specific heat of Fe60Ni20B10Si10 bulk metallic glass using HB approach.

152

Aditya M. Vora

2

TG

Fe60Ni20B10Si20

-2

2

CV/T (10 J/(mol-K ))

1.5

1

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0.5 Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 7. Dependence on screening on low temperature specific heat of Fe60Ni20B10Si10 bulk metallic glass using TG approach.

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

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0.15

BS

2

CV/T (10 J/(mol-K ))

0.12

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-2

0.09

0.06

0.03

Hartree Taylor Ichimaru-Utsumi Farid et al. Sarkar et al.

Fe60Ni20B10Si20 0 0

20

40 2

60 2

80

100

2

T (10 K ) Figure 8. Dependence on screening on low temperature specific heat of Fe60Ni20B10Si10 bulk metallic glass using BS approach.

The thermodynamic and elastic properties estimated from the elastic limit of the PDC are narrated in Table 2. It is seen that the results due to HB and TG approaches are very low in comparison with BS approach. The effect of various local field correction functions is also distinguishable on the thermodynamic and elastic properties of Fe60Ni20B10Si10 glass.

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Table 2. Thermodynamic and Elastic properties of Fe60Ni20B10Si10 bulk metallic glass

υL x App.

HB

TG

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BS

SCR H T IU F S H T IU F S H T IU F S

5

10 cm/s 2.35 2.79 2.89 2.91 1.70 5.04 5.76 5.71 5.81 4.97 10.07 10.22 10.21 10.23 10.20

υT x 5

10 cm/s 1.36 1.61 1.67 1.68 0.98 2.82 3.38 3.31 3.37 2.89 2.39 2.64 2.62 2.65 2.61

BT x 1011

G x 1011

dyne/cm2

dyne/cm

2.02 2.86 3.06 3.10 1.06 9.76 11.85 11.87 12.27 8.93 61.88 62.76 62.65 62.78 62.60

1.21 1.71 1.83 1.86 0.63 5.23 7.51 7.22 7.50 5.51 3.76 4.58 4.53 4.62 4.49

2

σ 0.25 0.25 0.25 0.25 0.25 0.27 0.24 0.25 0.25 0.24 0.47 0.46 0.46 0.46 0.47

Y x 1011 dyne/cm2

θD (K)

3.03 4.28 4.59 4.66 1.58 13.31 18.61 18.00 18.69 13.72 11.06 13.42 13.27 13.52 13.16

188.27 223.83 231.60 233.36 136.09 392.04 468.08 459.20 468.03 401.22 341.21 376.37 374.24 377.80 372.65

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 some bulk metallic glass. The influences of various local field correction functions are also observed in the present study. The experimental or theoretical data of most of the bulk metallic glass are not available in the literature but the present study is very useful to form a set of theoretical data of particular bulk metallic glass. In all three approaches, it is very difficult to judge which approaches is best for computations of vibrational dynamics of bulk metallic glass, because each approximation has its own identity. The HB approach is simplest and older one, which generating consistent results of the vibrational data of these bulk metallic glass, because the HB approaches needs minimum number of parameters. While 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 is retained the interatomic interactions effective between the first nearest neighbours only hence, the disorderness of the atoms in the formation of metallic glasses is more which show deviation in magnitude of the PDC and their related properties. From the present study we are concluded that, all three approaches are suitable for studying the vibrational dynamics of the amorphous materials. Hence, successful application of the model potential with three approaches is observed from the present study. The dielectric function plays an important role in the evaluation of potential due to the screening of the electron gas. For this purpose in the present investigations the local filed correction function due to H, T, IU, F and S are used. Reason for selecting these functions is that H-function does not include exchange and correlation effect and represents only static dielectric function, while T-function cover the overall features of the various local field correction functions proposed before 1972. While, IU, F and S functions are recent one among the existing functions and not exploited rigorously in such study. This helps us to

Study of Vibrational Dynamics of Fe60Ni20B10Si10 Bulk Metallic Glass

155

study the relative effects of exchange and correlation in the aforesaid properties. Hence, the five different local field correction functions show variations up to an order of magnitude in the vibrational properties.

CONCLUSIONS Finally, it is concluded that the PDC generated form three approaches with five local field correction functions reproduce all broad characteristics of dispersion curves. The well recognized model potential with more advanced IU, F and S-local field correction functions generate consistent results. The experimentally or theoretically observed data of most of the bulk metallic glass are not available in the literature. Therefore, it is difficult to draw any special remarks. However, the present study is very useful to provide important information regarding the particular glass. Also, the present computation confirms the applicability of the model potential in the aforesaid properties and supports the present approach of PAA. Such study on phonon dynamics of other binary liquid alloys and bulk metallic glasses is in progress.

REFERENCES

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Vora Aditya M. Ph. D. Thesis; Sardar Patel University: INDIA, 2004. Gajjar P. N.; Vora A. M.; Jani A. R. Proc. 9th Asia Pacific Phys. Conf., The Gioi Publ.: Vietnam, 2006, pp 429-433. Vora A. M.; Patel M. H.; Gajjar P. N.; Jani A. R., Solid State Phys. 2003, 46, 315-316. Vora Aditya M. Chinese Phys. Lett. 2006, Vol. 23, pp 1872-1875. Vora Aditya M. J. Non-Cryst. Sol. 2006, Vol. 52, 3217-3223. Vora Aditya M. J. Mater. Sci. 2007, Vol. 43, 935-940. Vora Aditya M. Acta Phys. Polo. A. 2007, Vol. 111, 859-871. Vora Aditya M. Front. Mater. Sci. China 2007, Vol. 1, 366-378. Vora Aditya M. FIZIKA A. 2007, Vol. 16, pp 187-206. Vora Aditya M. Romanian J. Phys. 2008, Vol. , pp - (in press). Sulir L. D.; Pyka M. Acta Phys. Pol. A. 1983, Vol. 63, pp 747-754. Hubbard J.; Beeby J. L. J. Phys. C: Solid State Phys. 1969, Vol. 2, pp 556-571. Takeno S.; Goda M. Prog. Thero. Phys. 1971, Vol. 45, pp 331-352. Takeno S.; Goda M. Prog. Thero. Phys. 1972, Vol. 47, pp 790-806. Bhatia A. B.; Singh R. N. Phys. Rev B. 1985, Vol. 31, pp 4751-4758. Shukla M. M.; Campanha J. R. Acta Phys. Pol. A. 1998, Vol. 94, pp 655-660. Harrison W. A. Elementary Electronic Structure; World Scientific: Singapore, 1999. Taylor R. J. Phys. F: Met. Phys. 1978, Vol. 8, pp 1699-1702. Ichimaru S.; Utsumi K. Phys. Rev. B. 1981, Vol. 24, pp 7385-7388. Farid B.; Heine V.; Engel G.; Robertson I. J. Phys. Rev. B. 1993, Vol. 48, pp 1160211621. Sarkar A.; Sen D. S.; Haldar S.; Roy D. Mod. Phys. Lett. B. 1998, Vol. 12, pp 639-648. Wills J. M.; Harrison W. A. Phys. Rev B. 1983, Vol. 28, pp 4363-4373.

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[23] Faber T. E. Introduction to the Theory of Liquid Metals; Cambridge Uni. Press: London, 1972. [24] Kovalenko N. P.; Krasny Y. P. Physics B. 1990, Vol. 162, pp 115-121. [25] Bretonnet J. L.; Derouiche A. Phys. Rev. B. 1990, Vol. 43, pp 8924- 8929. [26] Thorpe M. F. J. Non-Cryst. Sol. 1983, Vol. 57, 355-370.

INDEX A  amorphous alloys, vii, 2, 37 amorphous metal, vii Asia, 33, 62, 83, 94, 113, 134, 155 atomic-scale structure, vii atoms, vii, 2, 22, 33, 52, 61, 72, 82, 92, 102, 112, 122, 133, 144, 154

B 

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Bhatia-Singh (BS), vii, viii, ix, 15, 16, 20, 21, 35, 36, 37, 45, 46, 50, 51, 65, 66, 70, 71, 85, 86, 95, 96, 100, 101, 115, 116, 120, 121, 137, 138, 142, 143 bulk metallic glasses (BMG), vii, 1, 2, 3, 6, 7, 9, 10, 11, 12, 13

C  candidates, 35 China, 33, 44, 62, 83, 113, 134, 155 combined effect, 24, 54, 73, 104, 123, 146 compressibility, 4, 5, 12, 17, 18, 47, 48, 60, 67, 68, 97, 98, 117, 118, 139, 140 computation, viii, 2, 11, 18, 20, 33, 37, 40, 48, 50, 62, 68, 70, 83, 85, 98, 100, 113, 118, 120, 134, 140, 142, 155 computing, 4, 16, 46, 66, 96, 116, 138 coordination, 19, 49, 69, 88, 99, 119, 141 correlation, vii, viii, ix, 2, 4, 12, 15, 16, 17, 20, 23, 24, 25, 27, 28, 32, 33, 35, 36, 37, 42, 45, 46, 47, 50, 53, 54, 56, 59, 60, 61, 65, 66, 67, 70, 72, 74, 75, 78, 81, 82, 85, 86, 87, 90, 95, 96, 97, 100, 102, 104, 105, 108, 112, 113, 115, 116, 117, 120, 123, 125, 126, 133, 134, 137, 138, 139, 142, 144, 146, 147, 150, 154 correlation function, vii, viii, ix, 2, 15, 16, 20, 24, 25, 28, 32, 35, 36, 37, 42, 45, 46, 50, 54, 59, 60, 61, 65, 66, 70, 74, 78, 82, 85, 86, 87, 95,

96, 100, 103, 104, 105, 108, 112, 115, 116, 120, 123, 125, 126, 133, 137, 138, 142, 146, 147, 150, 154 correlations, 5, 12, 18, 48, 68, 98, 118, 140 Coulomb interaction, 10 Coulomb pseudopotential, vii, 1, 2, 4, 10, 11, 13 crystalline, vii, 16, 26, 32, 42, 46, 55, 61, 66, 75, 82, 96, 105, 112, 116, 123, 133, 138, 146, 154

D  Debye temperature, viii, 3, 4, 16, 22, 36, 39, 46, 52, 66, 71, 85, 89, 96, 101, 116, 122, 138, 143 degenerate, 5, 12, 18, 47, 67, 98, 117, 139 depth, 23, 40, 53, 72, 103, 123, 145 deviation, 28, 33, 59, 61, 78, 82, 92, 108, 112, 126, 133, 150, 154 disorder, 20, 26, 37, 50, 55, 70, 75, 100, 105, 120, 123, 142, 146 dispersion, vii, viii, ix, 15, 16, 25, 26, 27, 28, 33, 35, 36, 43, 45, 46, 55, 56, 57, 62, 65, 66, 76, 77, 78, 83, 85, 88, 95, 96, 106, 107, 108, 113, 115, 116, 127, 128, 129, 134, 137, 138, 148, 149, 150, 155 displacement, 33, 61, 82, 112, 133, 154 dynamical properties, 35

E  Elastic properties, 15, 32, 35, 45, 61, 65, 82, 93, 95, 112, 115, 133, 137, 154 electron, vii, viii, ix, 1, 2, 3, 4, 5, 9, 11, 12, 13, 15, 16, 18, 19, 24, 33, 35, 36, 40, 45, 46, 47, 49, 54, 61, 65, 66, 67, 69, 72, 82, 85, 86, 90, 95, 96, 98, 99, 103, 104, 113, 115, 116, 117, 119, 123, 134, 137, 138, 139, 141, 145, 146, 154 electronic structure, 36 electron-phonon coupling, vii, 1, 2, 3, 9, 11, 12, 13

158

Index

electron-phonon coupling strength, vii, 1, 2, 3, 9, 11, 12, 13 electrons, 2, 10, 24, 54, 73, 104, 123, 146 empty core (EMC) model potential, vii, 1, 2, 3 energy, 4, 17, 23, 36, 40, 47, 53, 67, 72, 87, 90, 97, 103, 117, 123, 139, 145 excitation, 91

F  fluid, 17, 47, 67, 87, 97, 117, 139 force, 16, 32, 61, 82, 86, 112, 133, 154 formation, 33, 61, 82, 112, 133, 154 formula, 5, 12, 18, 19, 47, 49, 67, 69, 98, 99, 117, 119, 139, 141

G  glasses, iv, vii, 1, 2, 3, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 33, 35, 36, 37, 43, 45, 46, 49, 61, 62, 65, 66, 69, 82, 83, 85, 87, 95, 96, 99, 112, 113, 115, 116, 119, 133, 134, 137, 138, 141, 154, 155

H 

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heat capacity, 16, 22, 46, 52, 66, 72, 96, 102, 116, 122, 138, 144 history, 15, 45, 65, 95, 115, 137 Hubbard-Beeby (HB), vii, viii, ix, 15, 16, 20, 35, 36, 45, 46, 50, 65, 66, 70, 85, 86, 95, 96, 100, 115, 116, 120, 137, 138, 142 hybridization, 10

I  Ichimaru-Utsumi (IU), vii, 1, 2, 4, 5, 12, 16, 17, 18, 46, 47, 48, 66, 67, 68, 96, 97, 98, 116, 117, 118, 138, 139, 140 identity, 32, 61, 82, 112, 133, 154 India, 1, 15, 35, 44, 45, 65, 85, 95, 115, 137 isotope, vii, 1, 2, 5, 11, 13 isotope effect exponent, vii, 1, 2, 5, 11, 13

L  lead, 12 liquids, 5, 12, 18, 47, 67, 98, 117, 139 lying, 28, 56, 75, 91, 105, 126, 146

M  magnitude, 11, 13, 33, 61, 62, 82, 83, 92, 111, 112, 113, 133, 134, 154, 155 manufacturing, 15, 45, 65, 95, 115, 137

mass, 3, 4, 17, 21, 38, 47, 51, 67, 71, 88, 97, 101, 117, 121, 139, 143 materials, 2, 15, 33, 45, 61, 65, 82, 95, 113, 115, 133, 137, 154 media, 15, 45, 65, 95, 115, 137 metals, vii, 2, 10 models, 86 modifications, ix modulus, viii, 16, 21, 36, 39, 46, 51, 66, 71, 85, 89, 93, 96, 101, 116, 121, 138, 143 momentum, 28, 57, 75, 91, 106, 126, 147

N  novel materials, 2

O  orthogonality, 19, 49, 69, 99, 119, 141 oscillation, 40

P  PAA, viii, ix, 2, 6, 15, 16, 33, 35, 36, 43, 45, 46, 62, 65, 66, 83, 95, 96, 113, 115, 116, 134, 137, 138, 155 Pacific, 33, 62, 83, 94, 113, 134, 155 Pair potential, 15, 35, 45, 65, 95, 115, 137 phonon dispersion, vii, viii, ix, 15, 16, 25, 26, 27, 35, 36, 45, 46, 55, 56, 57, 65, 66, 76, 77, 78, 85, 95, 96, 106, 107, 108, 115, 116, 127, 128, 129, 137, 138, 148, 149, 150 Phonon dispersion curves, 15, 35, 45, 65, 95, 115, 137 phonon dispersion curves (PDC), 16, 36, 46, 66, 85, 96, 116, 138 phonon eigen frequencies, vii, viii, ix, 15, 27, 35, 45, 56, 65, 75, 85, 95, 105, 115, 126, 137, 146 phonons, 10, 28, 37, 57, 75, 87, 91, 106, 126, 147, 149 pseudo-alloy-atom (PAA) model, viii, ix, 2, 15, 35, 45, 65, 95, 115, 137 pseudopotential theory, 2, 3, 16, 27, 46, 66, 85, 96, 116, 138

R  radius, 19, 25, 42, 49, 55, 69, 74, 99, 105, 119, 123, 141, 146 repulsion, 19, 49, 69, 99, 119, 141 room temperature, vii, viii, ix, 15, 20, 35, 36, 37, 45, 50, 65, 70, 85, 95, 100, 115, 120, 137, 142 rules, 12

Index

S  shape, 24, 26, 54, 55, 72, 75, 103, 105, 123, 145, 146 showing, 11, 12 Singapore, 14, 34, 62, 83, 114, 134, 155 South Asia, 13 specific heat, 16, 28, 29, 30, 31, 46, 58, 59, 60, 66, 78, 79, 80, 81, 90, 92, 96, 108, 109, 110, 111, 116, 126, 129, 130, 131, 132, 138, 150, 151, 152, 153 state, vii, 1, 2, 6, 7, 12, 13, 19, 42, 49, 69, 99, 119, 141 states, 3, 17, 28, 47, 59, 67, 78, 91, 92, 97, 108, 117, 129, 139, 150 storage, 15, 45, 65, 95, 115, 137 storage media, 15, 45, 65, 95, 115, 137 structure, vii, 17, 20, 25, 28, 36, 41, 42, 47, 50, 55, 67, 69, 75, 91, 96, 97, 100, 105, 117, 120, 123, 139, 142, 146 superconducting state parameters (SSP), vii, 1, 2, 6, 12, 13 superconductivity, 2, 11, 13 superconductor, 12 suppression, 82

159 theoretical approaches, 16, 36, 42, 46, 66, 96, 116, 138 thermal properties, 85 Thermal properties, 15, 35, 45, 65, 95, 115, 137 thermodynamic properties, viii, ix, 15, 16, 35, 45, 46, 65, 66, 95, 96, 115, 116, 137, 138 thermodynamics, 43 tin, 19, 49, 69, 99, 119, 141 transition metal, 10, 19, 46, 49, 65, 69, 95, 99, 115, 119, 137, 141 transition temperature, vii, 1, 2, 5, 10, 11, 12, 13 transmission, 11 treatment, 3

V  valence, 3, 12, 17, 36, 47, 67, 87, 97, 117, 139 variations, 33, 62, 83, 113, 134, 155 vector, 3, 17, 20, 28, 36, 38, 47, 50, 58, 67, 70, 75, 91, 97, 100, 107, 117, 120, 126, 139, 142, 148 velocity, 16, 36, 46, 66, 96, 116, 138 vibrational dynamics, vii, viii, ix, 15, 16, 32, 35, 43, 45, 46, 61, 65, 66, 82, 95, 96, 112, 115, 116, 133, 137, 138, 154 Vietnam, 33, 62, 83, 94, 113, 134, 155

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T  Takeno-Goda (TG), vii, viii, ix, 15, 16, 20, 35, 36, 37, 45, 46, 50, 65, 66, 70, 85, 86, 95, 96, 100, 115, 116, 120, 137, 138, 142 temperature, vii, viii, ix, 1, 2, 3, 4, 5, 10, 11, 12, 13, 15, 16, 20, 22, 28, 29, 30, 31, 35, 36, 37, 39, 45, 46, 50, 52, 58, 59, 60, 65, 66, 70, 71, 72, 78, 79, 80, 81, 85, 89, 90, 92, 95, 96, 100, 101, 102, 109, 110, 111, 115, 116, 120, 122, 129, 130, 131, 132, 137, 138, 142, 143, 144, 150, 151, 152, 153 temperature dependence, 22, 28, 36, 52, 59, 72, 78, 102, 109, 122, 129, 144, 150

W  wave number, 17, 36, 47, 67, 86, 87, 97, 117, 139 wave vector, 3, 17, 20, 21, 28, 36, 38, 47, 50, 51, 58, 67, 70, 71, 75, 88, 91, 97, 100, 101, 107, 117, 120, 121, 126, 139, 142, 143, 148 Wills and Harrison (WH), 19, 49, 69, 99, 119, 141

Y  yield, 4