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Theoretical study of vibrational dynamics and properties of Cu57Zr43 metallic glass
Materials Science and Engineering A 375–377 (2004) 697–700
Theoretical study of vibrational dynamics and properties of Cu57Zr43 metallic glass Arun Pratap a,∗ , Deepika Bhandari b , N.S. Saxena c a
c
Applied Physics Department, Condensed Matter Physics Laboratory, Faculty of Technology & Engineering, M.S. University of Baroda, Vadodara-390 001, India b Department of Physics, S.S. Jain Subodh College, Jaipur-302 004, India Department of Physics, Condensed Matter Physics Laboratory, University of Rajasthan, Jaipur-302 004, India
Abstract Density fluctuations in Cu57 Zr43 metallic glass have been studied in terms of eigen-frequencies of longitudinal and transverse phonon modes in effective medium approximation. Two different theoretical approaches used for the numerical calculations are: (1) self-consistent phonon theory developed by Takeno and Goda for amorphous solids and (2) model approach of Hubbard and Beeby. The results have been compared with the results of Kobayashi and Takeuchi obtained through recursion method. The low wave vector transfer region of computed dispersion curves have been used in order to determine the elastic and thermal properties of the glass. The results are found to be in excellent agreement with the results obtained through recursion technique. Further, the vibrational part of specific heat using phonon eigen-frequencies is also calculated and compared with experimental results of Suck et al. A good qualitative agreement is found. © 2003 Elsevier B.V. All rights reserved. Keywords: Pair-potential; Dispersion curves; Debye temperature; Elastic properties
1. Introduction The knowledge of the atomic structure and vibrational dynamics on the basis of inter-atomic forces is required for the understanding of thermodynamics, transport and other properties of condensed systems on a microscopic level [1]. Several attempts have been made to construct model structures of binary alloys viz. metal–metalloid and metal–metal ones. But, stress has been given more on the former [2,3] and less on the latter [4–6]. The structure and thermo-dynamical properties of transition metal-based amorphous alloys still remain unclear up to a large extent both experimentally and theoretically. One reason for this “retardation” in the investigation of the atomic structure of the glassy metals using neutron diffraction is the fact that it becomes very difficult to obtain a sufficient amount of glassy sample (10–50 g) required for neutron inelastic (NIS) experiments. Besides the experimental work, few computer simulations [3] and model calculations [7–9] have been performed to
∗
Corresponding author. E-mail address: [email protected] (A. Pratap).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.10.171
study the dynamics of these disordered systems. Recently, the structure of phonons in amorphous solids has also been investigated from the viewpoint of quantum field theory [10]. The purpose of this attempt is essentially the determination of longitudinal acoustic phonons, the phonon dispersion curve and lifetime of phonons, starting with atoms interacting with each other from first principles. The theoretical problem in the investigation of the structure and dynamics of transition metal-based amorphous alloys is the linear response method used to express the cohesive energy of a simple metal as sum of a volume term and of pair and higher order interaction can not be used for transition metals. These methods are applicable only to systems whose ions scatter the conduction electrons weakly and are describable by weak pseudo potentials. Serious efforts have been put in developing alternative methods which are also valid for strong-scattering open shell systems such as the transition metals and in this regard, recent work of Willis and Harrison [11] seems to have overcome most of the difficulties. In the present paper, a new effective pair-potential is proposed for Cu57 Zr43 glass in Willis–Harrison form treating the amorphous material as a pseudo one-component system of effective atoms. Two independent theories of phonons
698
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
in amorphous solids used quite extensively for disordered solids like liquid alloys [12] and metal–metal glasses [13] have been employed to compute the eigen-frequencies of longitudinal and transverse acoustic phonons with the so-obtained potential. The elastic constants and Debye temperature evaluated from the sound velocity derived from dispersion curves have been compared with the experimental results and recursion technique values [14]. The dispersion relations have been further used to study the temperature variation of the low-temperature heat capacity of the binary glassy system.
The pair-potential of transition metals in Wills and Harrison [11] form consists of an s-electron contribution Vs (r), a d-band bonding term Vb (r) and the repulsive contribution Vr (r) from the shift of the d-band centres. (1)
The s-electron contribution to the pair-potential is given by the expression [15]: 2Zs2 χ(q) 2 sin(qr) Vs (r) = 1 + 16 cos (qrc ) dq (2) r ε(q) q3 where Zs is the number of s-electrons and rc is the empty-core radius. The other terms have their usual meaning and have been discussed in detail elsewhere [16]. The d-electron contribution are expressed in terms of the number of d-electrons Zd , the d-state radii rd and the nearest-neighbour co-ordination number Nc as 1/2 1 − Zd 12 28.06 2rd3 (3) Vb (r) = −Zd 10 Nc π r5 450 rd6 (4) Vr (r) = Zd π2 r8 The term Vb (r) takes into account the Friedel-model band broadening contribution to the transition metal cohesion, while Vr (r) arises from the repulsion of the d-electron muffin-tin orbitals on different sites due to their non-orthogonality. Wills and Harrison have studied the effect of s–d hybridisation by changing the relative occupancy of the s and d bands. Zs , Zd and rd are determined from the band structure data. The effective pair-potential in binary transition-metal glass can also be obtained by treating the system as made up of effective atoms of mass: M = xMA + (1 − x)MB
(5)
and effective number density, ρ = xρA + (1 − x)ρB
Zdeff = (Zd )A CA + (Zd )B CB
(7)
being the effective d-electron valency. The d-electron radius of the effective medium is obtained by the relation
2. Theory
V(r) = Vs (r) + Vb (r) + Vr (r)
The s-electron contribution has its usual expression with modified parameters to take into account the interaction between effective atoms. This equation has been successfully applied to simple s–p bonded binary systems [16,17].In the same spirit, the other two contributions i.e. Friedel-model band broadening and the repulsive contribution from the shift of d-band centres, are used with modified parameters for d-electrons of effective pair of atoms. The parameter Zd is modified to Zdeff , treating the system with a pseudo one-component model with
(6)
where MA , MB, ρA and ρB are the mass and number density of A and B components respectively.
x 43 π((rd )A )3 + (1 − x) 43 π((rd )B )3 = 43 π((rd )eff )3
(8)
where (rd )A , (rd )B and (rd )eff are the d-electron radii of A, B and effective atom respectively. The phonon eigen-frequencies are physically meaningful quantities with which we study the an-harmonicity of the system. Hubbard and Beeby [12] introduced a theory to study the collective motions of one-component liquids. This theory has been successfully applied to two-component systems viz. liquid alloys, metallic glasses [17] and super-ionic solids [18]. The expressions for both the longitudinal and transverse phonon frequencies are given as 3 sin(qσ) 6 cos(qσ) 6 sin(qσ) 2 2 ωl (q) = ωE 1 − − (9) + qσ (qσ)2 (qσ)3 3 cos(qσ) 3 sin(qσ) ωt2 (q) = ωE2 1 − (10) + (qσ)2 (qσ)3 where ωE =
4πρ M
1/2
dr r2 g(r)V (r)
(11)
and σ are the maximum frequency and hard-core diameter respectively. The theory proposed by Takeno [19] for the lattice dynamics of an-harmonic solids from the viewpoint of self-consistent phonon theory was extended by Takeno and Goda to study the phonons in non-crystalline solids and liquids [13]. Grand success of this method in ordinary binary alloys and glasses in our earlier work [6,20,21] tempted us to use it in binary transition metallic glasses. The longitudinal and transverse frequencies in terms of effective pair correlation function and derivatives of the potentials are givens as ∞ 4πρ sin(qr) 2 ωl (q) = dr g(r) rV (r) 1 − M qr 0 + {r2 V (r) − rV (r)} 1 sin(qr) cos(qr) sin(qr) × − −2 + 2 3 qr (qr)2 (qr)3
(12)
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
Fig. 1. Effective pair potential of Cu57 Zr43 metallic glass computed using Eq. (1).
and
ωt2 (q) =
4πρ M
0
∞
sin(qr) dr g(r) rV (r) 1 − qr
+ {r2 V (r) − rV (r)} 1 cos(qr) sin(qr) + × − 3 (qr)2 (qr)3
(13)
respectively. The distinctive features of the temperature dependence of the heat capacity Cv (T) of the metallic glass are determined by the behaviour of ω (q) using the expression [22]; Vh ¯2 CV = kB T 2 λ
×
dq (2π)3
ωλ2 (q) (exp(¯hωλ (q)/kB T) − 1)(1 − exp(−¯hωλ (q)/kB T)) (14)
3. Results and discussion The effective pair-potential using the concept of Wills–Harrison form is computed employing Eq. (1). The computed effective pair-potential is shown in Fig. 1. It can be noted that this potential does not show any oscillations at larger r values.
699
Fig. 2. The longitudinal (L) and transverse (T) phonon eigen-frequencies for Cu57 Zr43 metallic glass (—): using Takeno and Goda approach; ): using Hubbard and Beeby approach; ( ( ): points of longi): points of transverse tudinal branch using recursion method [14]; ( branch using recursion method [14].
Using the effective pair-potential, the phonon eigenfrequencies of longitudinal and transverse modes have been calculated employing the two different approaches given by Hubbard and Beeby [12] and Takeno and Goda [13]. The values of effective pair correlation function have been taken from neutron diffraction results of Lamparter et al [23]. The dispersion curves are displayed in Fig. 2. The other theoretical results using the recursion method are also in the same figure for comparison. It can be seen from Fig. 2 that both curves reproduce all the characteristic features of the dispersion relations. The first minima in the longitudinal branch (computed using the theory of Hubbard and Beeby) lies at the same value of wave vector transfer, where the first peak of the static structure factor S(q) [23] occurs. On comparison of our computed results with the results obtained through recursion method, it can be noted that there is quantitative agreement between the two results. However, our computed results lie above the results obtained by recursion method [14]. From the elastic part of the dispersion curves, the sound velocities of longitudinal and transverse branch have been estimated. Using the values of longitudinal and transverse velocities, the elastic isothermal bulk modulus and Debye temperature have been calculated and compared with the crystalline state. These properties calculated are listed in Table 1 along with the values obtained through recursion method [14]. It can be visualised from the table that the
Table 1 Values of thermodynamic and elastic properties for Cu57 Zr43 glass
Takeno and Goda [13] Hubbard and Beeby [12] Kobayashi and Takeuchi [14]
Cl (105 cm/s)
Ct (105 cm/s)
ΘD (K)
BT (1011 dyne/cm2 )
4.36 4.01 4.02
2.56 2.59 2.06
331.59 332.26 327.00
8.10 5.59 –
700
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
shows close agreement with the much sophisticated recursion technique results. This indicates that the actual inter-atomic interactions in the system are quite close to the one incorporated in deriving the dispersion curves in the present work in Wills and Harrison form. The elastic and thermal properties derived from the curves are in fair match with those obtained from recursion technique. Further, the vibrational part of low temperature specific heat shows the temperature variation as observed by Suck et al. References
Fig. 3. Temperature dependence of the vibrational part of the specific heat of Cu57 Zr43 metallic glass (—): present computed results; (×): results of Suck and Rudin [24].
values of our computed results agree well with the values obtained through recursion method. Further, the values of temperature dependent specific heat are also computed using the Eq. (14). The values of Cv /T are plotted in Fig. 3 as a function of T2 . The experimental values of the vibrational part of the specific heat capacity [24] is also shown in the same figure. On comparison, we find that the two results show the usual anomalous behaviour and differ from the well-known T3 law in the low temperature region and agree qualitatively.
4. Conclusion The present work obtains pair-potential for a binary transition metal amorphous alloy viz. Cu57 Zr43 treating it as a pseudo one-component system of effective atoms. The assumption may, at the first sight, appear to be over simplified. But, the collective excitations in terms of dispersion relations derived from so-obtained effective pair-potential
[1] J.B. Suck, KFK report (1975) 2231. [2] S.C. Moss, D.L. Rice, J.M. Carpenter, D. Pan, D. Turnbull, Bull. Am. Phys. Soc. 19 (1974) 321. [3] Y. Ishi, T. Fujiwara, J. Phys. F 10 (1980) 2125. [4] J. Hafner, Phys. Rev. B 27 (1982) 678. [5] J.B. Suck, H. Rudin, H.J. Guntherodt, H. Beck, J. Phys. C 13 (1980) L1045. [6] N.S. Saxena, D. Bhandari, A. Pratap, M.P. Saksena, J. Phys.: Condens. Matter 2 (1990) 9475. [7] L.J. Lewis, N.W. Ashcroft, Phys. Rev. B 34 (1986) 8477. [8] Y.R. Wang, A.W. Overhauser, Phys. Rev. B 38 (1988) 9601. [9] N.S. Saxena, A. Pratap, D. Bhandari, M.P. Saksena, Mater. Sci. Eng. A134 (1991) 927. [10] T. Kitamura, Phys. Lett. A 152 (1991) 218. [11] J.M. Wills, W.A. Harrison, Phys. Rev. B 28 (1983) 4363. [12] J. Hubbard, J.L. Beeby, J. Phys. C 2 (1968) 556. [13] S. Takeno, M. Goda, Prog. Theor. Phys. 45 (1971) 331. [14] S. Kobayashi, S. Takeuchi, J. Phys. C 13 (1980) L969. [15] J. Hafner, From Hamiltonians to phase diagrams–the electronic and statistical–mechanical theory of sp-bonded metals and alloys, Solid State Science Series 70, Springer, Berlin, 1987. [16] D. Bhandari, A. Pratap, N.S. Saxena, M.P. Saxena, Phys. Status Sol. B 160 (1990) 83. [17] N.S. Saxena, M. Rani, A. Pratap, Prabhuram, M.P. Saksena, Phys. Rev. B 38 (1988) 8093. [18] S. Tomizawa, T. Masumoto, Sci. Rep. Res. Inst. Tohoku Univ A 26 (1976) 263. [19] S. Takeno, Prog. Theor. Phys. Suppl. 45 (1970). [20] K.C. Jain, N. Gupta, N.S. Saxena, J. Phys. Condens. Matter 3 (1991) 7485. [21] A. Pratap, D. Bhandari, N.S. Saxena, M.P. Saxena, Proc. Natl. Acad. Sci. India 62A (1992) 285. [22] N.P. Kovalenko, Yu.P. Krasnyi, V.N. Makhlaichuk, V.F. Tsarev, Sov. J. Low. Temp. Phys. 13 (1987) 535. [23] P. Lamparter, S. Steeb, E. Grallath, Z. Naturforsch 38 (1983) 1210. [24] J. B. Suck, H. Rudin, in: H. Beck, H.J. Guntherodt (Eds.) Glassy Metals. Part II. Topics in Applied Physics, Vol. 53, Springer-Verlag, Heidelberg, 1983, p. 217.
Theoretical study of vibrational dynamics and properties of Cu57Zr43 metallic glass Arun Pratap a,∗ , Deepika Bhandari b , N.S. Saxena c a
c
Applied Physics Department, Condensed Matter Physics Laboratory, Faculty of Technology & Engineering, M.S. University of Baroda, Vadodara-390 001, India b Department of Physics, S.S. Jain Subodh College, Jaipur-302 004, India Department of Physics, Condensed Matter Physics Laboratory, University of Rajasthan, Jaipur-302 004, India
Abstract Density fluctuations in Cu57 Zr43 metallic glass have been studied in terms of eigen-frequencies of longitudinal and transverse phonon modes in effective medium approximation. Two different theoretical approaches used for the numerical calculations are: (1) self-consistent phonon theory developed by Takeno and Goda for amorphous solids and (2) model approach of Hubbard and Beeby. The results have been compared with the results of Kobayashi and Takeuchi obtained through recursion method. The low wave vector transfer region of computed dispersion curves have been used in order to determine the elastic and thermal properties of the glass. The results are found to be in excellent agreement with the results obtained through recursion technique. Further, the vibrational part of specific heat using phonon eigen-frequencies is also calculated and compared with experimental results of Suck et al. A good qualitative agreement is found. © 2003 Elsevier B.V. All rights reserved. Keywords: Pair-potential; Dispersion curves; Debye temperature; Elastic properties
1. Introduction The knowledge of the atomic structure and vibrational dynamics on the basis of inter-atomic forces is required for the understanding of thermodynamics, transport and other properties of condensed systems on a microscopic level [1]. Several attempts have been made to construct model structures of binary alloys viz. metal–metalloid and metal–metal ones. But, stress has been given more on the former [2,3] and less on the latter [4–6]. The structure and thermo-dynamical properties of transition metal-based amorphous alloys still remain unclear up to a large extent both experimentally and theoretically. One reason for this “retardation” in the investigation of the atomic structure of the glassy metals using neutron diffraction is the fact that it becomes very difficult to obtain a sufficient amount of glassy sample (10–50 g) required for neutron inelastic (NIS) experiments. Besides the experimental work, few computer simulations [3] and model calculations [7–9] have been performed to
∗
Corresponding author. E-mail address: [email protected] (A. Pratap).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.10.171
study the dynamics of these disordered systems. Recently, the structure of phonons in amorphous solids has also been investigated from the viewpoint of quantum field theory [10]. The purpose of this attempt is essentially the determination of longitudinal acoustic phonons, the phonon dispersion curve and lifetime of phonons, starting with atoms interacting with each other from first principles. The theoretical problem in the investigation of the structure and dynamics of transition metal-based amorphous alloys is the linear response method used to express the cohesive energy of a simple metal as sum of a volume term and of pair and higher order interaction can not be used for transition metals. These methods are applicable only to systems whose ions scatter the conduction electrons weakly and are describable by weak pseudo potentials. Serious efforts have been put in developing alternative methods which are also valid for strong-scattering open shell systems such as the transition metals and in this regard, recent work of Willis and Harrison [11] seems to have overcome most of the difficulties. In the present paper, a new effective pair-potential is proposed for Cu57 Zr43 glass in Willis–Harrison form treating the amorphous material as a pseudo one-component system of effective atoms. Two independent theories of phonons
698
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
in amorphous solids used quite extensively for disordered solids like liquid alloys [12] and metal–metal glasses [13] have been employed to compute the eigen-frequencies of longitudinal and transverse acoustic phonons with the so-obtained potential. The elastic constants and Debye temperature evaluated from the sound velocity derived from dispersion curves have been compared with the experimental results and recursion technique values [14]. The dispersion relations have been further used to study the temperature variation of the low-temperature heat capacity of the binary glassy system.
The pair-potential of transition metals in Wills and Harrison [11] form consists of an s-electron contribution Vs (r), a d-band bonding term Vb (r) and the repulsive contribution Vr (r) from the shift of the d-band centres. (1)
The s-electron contribution to the pair-potential is given by the expression [15]: 2Zs2 χ(q) 2 sin(qr) Vs (r) = 1 + 16 cos (qrc ) dq (2) r ε(q) q3 where Zs is the number of s-electrons and rc is the empty-core radius. The other terms have their usual meaning and have been discussed in detail elsewhere [16]. The d-electron contribution are expressed in terms of the number of d-electrons Zd , the d-state radii rd and the nearest-neighbour co-ordination number Nc as 1/2 1 − Zd 12 28.06 2rd3 (3) Vb (r) = −Zd 10 Nc π r5 450 rd6 (4) Vr (r) = Zd π2 r8 The term Vb (r) takes into account the Friedel-model band broadening contribution to the transition metal cohesion, while Vr (r) arises from the repulsion of the d-electron muffin-tin orbitals on different sites due to their non-orthogonality. Wills and Harrison have studied the effect of s–d hybridisation by changing the relative occupancy of the s and d bands. Zs , Zd and rd are determined from the band structure data. The effective pair-potential in binary transition-metal glass can also be obtained by treating the system as made up of effective atoms of mass: M = xMA + (1 − x)MB
(5)
and effective number density, ρ = xρA + (1 − x)ρB
Zdeff = (Zd )A CA + (Zd )B CB
(7)
being the effective d-electron valency. The d-electron radius of the effective medium is obtained by the relation
2. Theory
V(r) = Vs (r) + Vb (r) + Vr (r)
The s-electron contribution has its usual expression with modified parameters to take into account the interaction between effective atoms. This equation has been successfully applied to simple s–p bonded binary systems [16,17].In the same spirit, the other two contributions i.e. Friedel-model band broadening and the repulsive contribution from the shift of d-band centres, are used with modified parameters for d-electrons of effective pair of atoms. The parameter Zd is modified to Zdeff , treating the system with a pseudo one-component model with
(6)
where MA , MB, ρA and ρB are the mass and number density of A and B components respectively.
x 43 π((rd )A )3 + (1 − x) 43 π((rd )B )3 = 43 π((rd )eff )3
(8)
where (rd )A , (rd )B and (rd )eff are the d-electron radii of A, B and effective atom respectively. The phonon eigen-frequencies are physically meaningful quantities with which we study the an-harmonicity of the system. Hubbard and Beeby [12] introduced a theory to study the collective motions of one-component liquids. This theory has been successfully applied to two-component systems viz. liquid alloys, metallic glasses [17] and super-ionic solids [18]. The expressions for both the longitudinal and transverse phonon frequencies are given as 3 sin(qσ) 6 cos(qσ) 6 sin(qσ) 2 2 ωl (q) = ωE 1 − − (9) + qσ (qσ)2 (qσ)3 3 cos(qσ) 3 sin(qσ) ωt2 (q) = ωE2 1 − (10) + (qσ)2 (qσ)3 where ωE =
4πρ M
1/2
dr r2 g(r)V (r)
(11)
and σ are the maximum frequency and hard-core diameter respectively. The theory proposed by Takeno [19] for the lattice dynamics of an-harmonic solids from the viewpoint of self-consistent phonon theory was extended by Takeno and Goda to study the phonons in non-crystalline solids and liquids [13]. Grand success of this method in ordinary binary alloys and glasses in our earlier work [6,20,21] tempted us to use it in binary transition metallic glasses. The longitudinal and transverse frequencies in terms of effective pair correlation function and derivatives of the potentials are givens as ∞ 4πρ sin(qr) 2 ωl (q) = dr g(r) rV (r) 1 − M qr 0 + {r2 V (r) − rV (r)} 1 sin(qr) cos(qr) sin(qr) × − −2 + 2 3 qr (qr)2 (qr)3
(12)
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
Fig. 1. Effective pair potential of Cu57 Zr43 metallic glass computed using Eq. (1).
and
ωt2 (q) =
4πρ M
0
∞
sin(qr) dr g(r) rV (r) 1 − qr
+ {r2 V (r) − rV (r)} 1 cos(qr) sin(qr) + × − 3 (qr)2 (qr)3
(13)
respectively. The distinctive features of the temperature dependence of the heat capacity Cv (T) of the metallic glass are determined by the behaviour of ω (q) using the expression [22]; Vh ¯2 CV = kB T 2 λ
×
dq (2π)3
ωλ2 (q) (exp(¯hωλ (q)/kB T) − 1)(1 − exp(−¯hωλ (q)/kB T)) (14)
3. Results and discussion The effective pair-potential using the concept of Wills–Harrison form is computed employing Eq. (1). The computed effective pair-potential is shown in Fig. 1. It can be noted that this potential does not show any oscillations at larger r values.
699
Fig. 2. The longitudinal (L) and transverse (T) phonon eigen-frequencies for Cu57 Zr43 metallic glass (—): using Takeno and Goda approach; ): using Hubbard and Beeby approach; ( ( ): points of longi): points of transverse tudinal branch using recursion method [14]; ( branch using recursion method [14].
Using the effective pair-potential, the phonon eigenfrequencies of longitudinal and transverse modes have been calculated employing the two different approaches given by Hubbard and Beeby [12] and Takeno and Goda [13]. The values of effective pair correlation function have been taken from neutron diffraction results of Lamparter et al [23]. The dispersion curves are displayed in Fig. 2. The other theoretical results using the recursion method are also in the same figure for comparison. It can be seen from Fig. 2 that both curves reproduce all the characteristic features of the dispersion relations. The first minima in the longitudinal branch (computed using the theory of Hubbard and Beeby) lies at the same value of wave vector transfer, where the first peak of the static structure factor S(q) [23] occurs. On comparison of our computed results with the results obtained through recursion method, it can be noted that there is quantitative agreement between the two results. However, our computed results lie above the results obtained by recursion method [14]. From the elastic part of the dispersion curves, the sound velocities of longitudinal and transverse branch have been estimated. Using the values of longitudinal and transverse velocities, the elastic isothermal bulk modulus and Debye temperature have been calculated and compared with the crystalline state. These properties calculated are listed in Table 1 along with the values obtained through recursion method [14]. It can be visualised from the table that the
Table 1 Values of thermodynamic and elastic properties for Cu57 Zr43 glass
Takeno and Goda [13] Hubbard and Beeby [12] Kobayashi and Takeuchi [14]
Cl (105 cm/s)
Ct (105 cm/s)
ΘD (K)
BT (1011 dyne/cm2 )
4.36 4.01 4.02
2.56 2.59 2.06
331.59 332.26 327.00
8.10 5.59 –
700
A. Pratap et al. / Materials Science and Engineering A 375–377 (2004) 697–700
shows close agreement with the much sophisticated recursion technique results. This indicates that the actual inter-atomic interactions in the system are quite close to the one incorporated in deriving the dispersion curves in the present work in Wills and Harrison form. The elastic and thermal properties derived from the curves are in fair match with those obtained from recursion technique. Further, the vibrational part of low temperature specific heat shows the temperature variation as observed by Suck et al. References
Fig. 3. Temperature dependence of the vibrational part of the specific heat of Cu57 Zr43 metallic glass (—): present computed results; (×): results of Suck and Rudin [24].
values of our computed results agree well with the values obtained through recursion method. Further, the values of temperature dependent specific heat are also computed using the Eq. (14). The values of Cv /T are plotted in Fig. 3 as a function of T2 . The experimental values of the vibrational part of the specific heat capacity [24] is also shown in the same figure. On comparison, we find that the two results show the usual anomalous behaviour and differ from the well-known T3 law in the low temperature region and agree qualitatively.
4. Conclusion The present work obtains pair-potential for a binary transition metal amorphous alloy viz. Cu57 Zr43 treating it as a pseudo one-component system of effective atoms. The assumption may, at the first sight, appear to be over simplified. But, the collective excitations in terms of dispersion relations derived from so-obtained effective pair-potential
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