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Structure and vibrational dynamics of Ca70Mg30 glass
Materials"Science and Engineering, A 134 ( 1991) 927- 930
927
Structure and vibrational dynamics of Cav0Mg30 glass N. S. Saxena, Arun Pratap, Deepika Bhandari and M. P. Saksena ( "ondensed ~latterI'hysics Laboratoo,, Department of Physics, UniversityofRajasthan, Jaipur-3020(14(India)
Abstract A new effective pair potential is proposed for CaToMg30glass in Ashcrofl form using the concept of a Wigner-Seitz sphere treating it as a one-component system. A theory of phonon in amorphous solids in a generalized random phase approximation is employed to compute the eigenfrequencies of the longitudinal and transverse phonons making use of the so obtained potential. The eigenfrequencies obtained agree qualitatively with the neutron inelastic scattering results of Suck et al. and also with the theoretical results of Harrier and with the results of Bhatia and Singh, both qualitatively as well as quantitatively. In addition, the elastic and thermodynamic properties of the glass have also been studied using the longitudinal and transverse sound velocities.
1. Introduction
effective mass
A knowledge of the atomic structure and vibrational dynamics on the basis of realistic interatomic forces is fundamental to the understanding of the thermodynamic, mechanical and electronic transport properties of amorphous materials [1-3]. Progress on the theoretical description has been hampered owing to the fact that metallic glasses show a larger degree of quantitative disorder, and consequently a more accurate knowledge of the interatomic force is required. The interatomic potential, using the pseudopotential theory, has already been found to be reliable to study the structure and dynamics of liquid metals [4, 5] and alloys [6, 7j. Owing to the stimulations by the measurements and studies done so far, in this paper we propose a new effective pair potential for Ca70Mg30 glass in the Ashcroft pseudopotential [8] form using the concept of a Wigner-Seitz sphere treating it as a one-component system. A theory of phonons in amorphous solids in a generalized random phase approximation (RPA) is employed to study the vibrational dynamics along with the elastic and thermodynamic properties of this glass [9].
M J * = X M a +(1 -X)MI~
2. Theory A simple metallic glass AxBl_ x can be looked upon as a one-component metallic fluid with 0921-5093/91/$3.50
and effective number density
p~ef= Xp~ + ( 1 - X)pB Here the subscripts A and B stand for the parameters of the two individual components of glass. The effective pair potential of the glass may be written in the usual form as
- 2(Z~%):/:t f dQ[sin(Qr)/Qr]F(Q)
(1)
where 1 )- 1 F(O)=[g2Oe/S:zee][ W,b(O)l , ( / ( Q and Z ~'rf= ZaQ ~+Z~ Cr~ is the effective valence of this glass. Qt and (~ are the concentrations of A and B, respectively [7]. The Fourier transform of the Ashcroft [8] pseudopotential is Wb(Q)= -[4:rZ~ffee/~Q2]cos(Qr~? "11)
(2)
and the dielectric response function e(Q) is given as
e(Q) = 1 -[{(4:re2/Q'-)z( Q)}/{ 1 + (4:ree/Q e)
×x(O)G(O)}]
(3)
© ElsevierSequoia/Printed in The Netherlands
928 Here G(Q) is called the "local field function" and is due to Hubbard and Sham [10] G(Q) = Qz/[2(Q2 + 'lJk~eff)) ]
(4)
where v = 2/(1 + 0.153/0"gkv(eff))
where we = [(4az/geff/3Meff)fdr r2geff(r)V~ff(r)]1/2 and o are the maximum frequency and hard core diameter, respectively. The hard core diameter of the glass is obtained as [12]
and
T] = ~]A-Jr-7]B
kF(eff) = ( 3 ~2 z efftoeff )l/3
where
rc°ff is called the ionic core radius of the bimetallic glass and is a disposable parameter. It is obtained using the relation
~]i= ~ ¢Ti ni
X.4~ra3+(l_X)4nrB3_4
with oi and n i being the hard core diameters and number density in the glass, respectively.
-
3 ~ :rrgtass
(5)
where rm, re and rgla+s are respectively the Wigner-Seitz radii of A, B and the glass. The Wigner-Seitz radius of the glass is related [11] to the effective core radius by the equation rc~ff= [0.51Fglass/(Zeff)1/3] The phonon eigenfrequencies are meaningful quantities to study the collective excitations in the system. The expressions for both the longitudinal and transverse phonon eigenfrequencies in a random phase approximation (RPA) are given as [9] w~2(Q) = we2[1 - 3 sin( Qo)/(Qo) - 6 cos(Qo)/(Qo) 2 + 8 sin( Qo)/(QÙ)3] (6) and wt2(Q) = we2[1 + 3 cos( Qcr)/(Qo) 2 - 3 sin( Qo)/(Qo) 3]
(7)
B 0
li // -16
-24 -32 Fig. 1. Effective pair potential of Ca70Mg30 glass: present calculations; - - - computed from Hafner's results
[131.
gl~
3
i= A , B
3. P h o n o n s in Ca70Mg30 glass
The new effective pair potential of CaToMg30 glass is computed with the help of eqn. (1) using the concept of Wigner-Seitz sphere in the Ashcroft form. This potential is plotted in Fig. 1. Hafner [13] has given the pair potentials for Ca-Ca, Mg-Mg and Ca-Mg. Using these individual potentials we have obtained the effective pair potential for the glass by the method used earlier [7] and this also has been plotted in Fig. 1. Both of the effective potentials are almost similar as far as the repulsive part is concerned. It can be observed from Fig. 1 that the present computed effective potential shows a slightly greater well depth moved towards the left as compared to the potential computed from the results of Hafner [13]. Moreover, the potential given by eqn. (1) shows significant oscillations and the potential energy remains positive in the larger r-region. It seems that the Coulomb repulsive potential part dominates the oscillations due to the ion-electron-ion interactions in this glass. Hence the present computed effective potential converges for r--, co towards a value greater than zero. The phonon eigenfrequencies for longitudinal and transverse modes calculated using eqns. (6) and (7) are shown in Fig. 2. The effective pair correlation function geff(r) is taken from the X-ray diffraction results of Nassif et al. [14]. The neutron scattering results of Suck et al. [15], the theoretical results of Bhatia and Singh [16] and of Hafner [13] are also shown in the same figure for comparison. The present results agree qualitatively well with the neutron inelastic scattering experimental points of Suck et al. [15] and results obtained by Bhatia and Singh [16]. Furthermore,
929
[17} 1 OD 2
/ ~/'
',
/
T 4>_,
J
,'
.
Q,~J"~ .-,
Fig. 2. Phonon dispersion curve of Cav0Mg~0 glass: - present results; @ (.D E) results of Hafner [13]; x x × neutron inelastic scattering results [14]; . . . . . results of Bhatia and 8ingh [15].
the present computed eigenfrequencies are in fair agreement both qualitatively as well as quantitatively with the theoretical results of Hafner [13]. It is obvious from Fig. 2 that the oscillations are prominent in the longitudinal p h o n o n m o d e as compared with the transverse one showing the collective excitations at larger wave vector transfers due to the dispersion of longitudinal phonons only. On the other hand, the transverse phonons undergo larger thermal modulation due to the anharmonicity of atomic vibrations in the glass. It may also be seen that the first minimum in the longitudinal branch of the dispersion curve falls at a value Q = 2 A- ~ close to Qp, i.e. the value of Q where the static structure factor S(Q) shows its first peak, T h e sharp first peak in the static structure factor indicates a certain degree of remnant long-range order. A density fluctuation with this wavelength produces only minimal atomic rearrangements and requires only a low excitation energy. In the long wavelength limit the dispersion curves are linear and have the characteristics of elastic waves. T h e values of the longitudinal p h o n o n velocity C 1 and the transverse p h o n o n velocity C~ derived from the elastic part of two p h o n o n modes are G = 5 . 6 6 x 1 0 5 cm s -1 and G = 3 . 5 5 x 105 cm s - L Using these values of C 1 and C~ and following Hafner [13] Debye temperature O D has been calculated for Cav0Mg30 glass. This result [O D = 347.63 K] has been compared with the value obtained through the expression
PA I - - -P8 ODA2 OD.2
and it deviates by 24%. H e r e PA and Pr~ are the concentration and OD, and ®D, are the Debye temperatures [18] of the pure components. Besides, the isothermal bulk modulus B T of an isotropic solid is given by B T = p ( G 2--4C]2/3), where p is the density of the isotropic solid. For amorphous Ca70Mg30, BT comes out to be 2.437 × 1011 dynes cm 2, which is within 4% of the averaged value for the crystalline metals (BT = 2.32 X 10 j~ dynes cm 2).
4. Conclusion It is found that the new effective pair potential together with the theory of phonons in amorphous solids in a generalized r a n d o m phase approximation gives a fairly accurate description of the vibrational dynamics as well as the thermal and elastic properties of Ca70Mg30 glass.
Acknowledgments Thanks are due to Mr. K. C. Jain and Ms. Neelam Gupta for their help in various ways.
References 1 J. Hafner, in H. Beck and H. J. Giintherodt (eds.), G l a s s y Metals L Topics in Applied Physics, Vol. 64, Springer,
Berlin, 1981, p. 93. 2 J. B. Suck and H. Rudin, in H. Beck and H. J. G6ntherodt (eds.), Glassy Metals" 11, Topics in Applied Physics, Vol. 53, Springer, Berlin, p. 217. 3 L. J, Lewis and N. W. Ashcroft, Phys. Rev., B34 (1986) 8477. 4 M. Rani, A. Pratap and N. S. Saxena, Phys. Status Solidi (b), 149(1988)93.
5 J.Hafner, J. Phys. F, 6 (1976) 1243. 6 J. Hafner. Phys. Rev. A, 16 (1977) 351. 7 Deepika Bhandari, Arun Pratap, N. S. Saxena and M. P. Saksena, Phys. Status Solidi ( b ), 160 (1990) 83. 8 N.W. Ashcroft, Phys. Lett., 23 (1966) 48. 9 J. Hubbard and L. Beeby, J. Phys. C, 2 (1969) 556. 10 J. Hubbard, Proc. R. Soc. London, A243 (1957) 336. L. J. Sham, Proc. R. Soc. London, A283 (1965) 33. 11 J. Hafner and V. Heine, J. Phys. F, 13 (1983) 2479. 12 Arun Pratap, Meeta Rani and N. S. Saxena, Pramana-J. Phys., 30 (1988) 239.
930 13 J. Hafner, Phys. Rev. B, 27(1983) 678. 14 E. Nassif, E Lamparter and S. Steeb, Z. Naturforsch A, 38 (1983) 1206. 15 J. B. Suck, H. Rudin, H. J. Giintherodt, D. Tomanek, H. Beck, C. Morkel and W. Gl~iser, J. Phys. (Paris') Colloq., 41 (1980) C8-175.
16 A. B. Bhatia and R. N. Singhl Phys. Rev. B, 31 (1985) 4751. 17 G. Grimval, Thermophysical Properties of Materials, North-Holland, Amsterdam, 1986. 18 N. W. Ashcroft and D. Stroud, Solid State Physics, 33 Academic Press, New York, 1977, p. 52.
927
Structure and vibrational dynamics of Cav0Mg30 glass N. S. Saxena, Arun Pratap, Deepika Bhandari and M. P. Saksena ( "ondensed ~latterI'hysics Laboratoo,, Department of Physics, UniversityofRajasthan, Jaipur-3020(14(India)
Abstract A new effective pair potential is proposed for CaToMg30glass in Ashcrofl form using the concept of a Wigner-Seitz sphere treating it as a one-component system. A theory of phonon in amorphous solids in a generalized random phase approximation is employed to compute the eigenfrequencies of the longitudinal and transverse phonons making use of the so obtained potential. The eigenfrequencies obtained agree qualitatively with the neutron inelastic scattering results of Suck et al. and also with the theoretical results of Harrier and with the results of Bhatia and Singh, both qualitatively as well as quantitatively. In addition, the elastic and thermodynamic properties of the glass have also been studied using the longitudinal and transverse sound velocities.
1. Introduction
effective mass
A knowledge of the atomic structure and vibrational dynamics on the basis of realistic interatomic forces is fundamental to the understanding of the thermodynamic, mechanical and electronic transport properties of amorphous materials [1-3]. Progress on the theoretical description has been hampered owing to the fact that metallic glasses show a larger degree of quantitative disorder, and consequently a more accurate knowledge of the interatomic force is required. The interatomic potential, using the pseudopotential theory, has already been found to be reliable to study the structure and dynamics of liquid metals [4, 5] and alloys [6, 7j. Owing to the stimulations by the measurements and studies done so far, in this paper we propose a new effective pair potential for Ca70Mg30 glass in the Ashcroft pseudopotential [8] form using the concept of a Wigner-Seitz sphere treating it as a one-component system. A theory of phonons in amorphous solids in a generalized random phase approximation (RPA) is employed to study the vibrational dynamics along with the elastic and thermodynamic properties of this glass [9].
M J * = X M a +(1 -X)MI~
2. Theory A simple metallic glass AxBl_ x can be looked upon as a one-component metallic fluid with 0921-5093/91/$3.50
and effective number density
p~ef= Xp~ + ( 1 - X)pB Here the subscripts A and B stand for the parameters of the two individual components of glass. The effective pair potential of the glass may be written in the usual form as
- 2(Z~%):/:t f dQ[sin(Qr)/Qr]F(Q)
(1)
where 1 )- 1 F(O)=[g2Oe/S:zee][ W,b(O)l , ( / ( Q and Z ~'rf= ZaQ ~+Z~ Cr~ is the effective valence of this glass. Qt and (~ are the concentrations of A and B, respectively [7]. The Fourier transform of the Ashcroft [8] pseudopotential is Wb(Q)= -[4:rZ~ffee/~Q2]cos(Qr~? "11)
(2)
and the dielectric response function e(Q) is given as
e(Q) = 1 -[{(4:re2/Q'-)z( Q)}/{ 1 + (4:ree/Q e)
×x(O)G(O)}]
(3)
© ElsevierSequoia/Printed in The Netherlands
928 Here G(Q) is called the "local field function" and is due to Hubbard and Sham [10] G(Q) = Qz/[2(Q2 + 'lJk~eff)) ]
(4)
where v = 2/(1 + 0.153/0"gkv(eff))
where we = [(4az/geff/3Meff)fdr r2geff(r)V~ff(r)]1/2 and o are the maximum frequency and hard core diameter, respectively. The hard core diameter of the glass is obtained as [12]
and
T] = ~]A-Jr-7]B
kF(eff) = ( 3 ~2 z efftoeff )l/3
where
rc°ff is called the ionic core radius of the bimetallic glass and is a disposable parameter. It is obtained using the relation
~]i= ~ ¢Ti ni
X.4~ra3+(l_X)4nrB3_4
with oi and n i being the hard core diameters and number density in the glass, respectively.
-
3 ~ :rrgtass
(5)
where rm, re and rgla+s are respectively the Wigner-Seitz radii of A, B and the glass. The Wigner-Seitz radius of the glass is related [11] to the effective core radius by the equation rc~ff= [0.51Fglass/(Zeff)1/3] The phonon eigenfrequencies are meaningful quantities to study the collective excitations in the system. The expressions for both the longitudinal and transverse phonon eigenfrequencies in a random phase approximation (RPA) are given as [9] w~2(Q) = we2[1 - 3 sin( Qo)/(Qo) - 6 cos(Qo)/(Qo) 2 + 8 sin( Qo)/(QÙ)3] (6) and wt2(Q) = we2[1 + 3 cos( Qcr)/(Qo) 2 - 3 sin( Qo)/(Qo) 3]
(7)
B 0
li // -16
-24 -32 Fig. 1. Effective pair potential of Ca70Mg30 glass: present calculations; - - - computed from Hafner's results
[131.
gl~
3
i= A , B
3. P h o n o n s in Ca70Mg30 glass
The new effective pair potential of CaToMg30 glass is computed with the help of eqn. (1) using the concept of Wigner-Seitz sphere in the Ashcroft form. This potential is plotted in Fig. 1. Hafner [13] has given the pair potentials for Ca-Ca, Mg-Mg and Ca-Mg. Using these individual potentials we have obtained the effective pair potential for the glass by the method used earlier [7] and this also has been plotted in Fig. 1. Both of the effective potentials are almost similar as far as the repulsive part is concerned. It can be observed from Fig. 1 that the present computed effective potential shows a slightly greater well depth moved towards the left as compared to the potential computed from the results of Hafner [13]. Moreover, the potential given by eqn. (1) shows significant oscillations and the potential energy remains positive in the larger r-region. It seems that the Coulomb repulsive potential part dominates the oscillations due to the ion-electron-ion interactions in this glass. Hence the present computed effective potential converges for r--, co towards a value greater than zero. The phonon eigenfrequencies for longitudinal and transverse modes calculated using eqns. (6) and (7) are shown in Fig. 2. The effective pair correlation function geff(r) is taken from the X-ray diffraction results of Nassif et al. [14]. The neutron scattering results of Suck et al. [15], the theoretical results of Bhatia and Singh [16] and of Hafner [13] are also shown in the same figure for comparison. The present results agree qualitatively well with the neutron inelastic scattering experimental points of Suck et al. [15] and results obtained by Bhatia and Singh [16]. Furthermore,
929
[17} 1 OD 2
/ ~/'
',
/
T 4>_,
J
,'
.
Q,~J"~ .-,
Fig. 2. Phonon dispersion curve of Cav0Mg~0 glass: - present results; @ (.D E) results of Hafner [13]; x x × neutron inelastic scattering results [14]; . . . . . results of Bhatia and 8ingh [15].
the present computed eigenfrequencies are in fair agreement both qualitatively as well as quantitatively with the theoretical results of Hafner [13]. It is obvious from Fig. 2 that the oscillations are prominent in the longitudinal p h o n o n m o d e as compared with the transverse one showing the collective excitations at larger wave vector transfers due to the dispersion of longitudinal phonons only. On the other hand, the transverse phonons undergo larger thermal modulation due to the anharmonicity of atomic vibrations in the glass. It may also be seen that the first minimum in the longitudinal branch of the dispersion curve falls at a value Q = 2 A- ~ close to Qp, i.e. the value of Q where the static structure factor S(Q) shows its first peak, T h e sharp first peak in the static structure factor indicates a certain degree of remnant long-range order. A density fluctuation with this wavelength produces only minimal atomic rearrangements and requires only a low excitation energy. In the long wavelength limit the dispersion curves are linear and have the characteristics of elastic waves. T h e values of the longitudinal p h o n o n velocity C 1 and the transverse p h o n o n velocity C~ derived from the elastic part of two p h o n o n modes are G = 5 . 6 6 x 1 0 5 cm s -1 and G = 3 . 5 5 x 105 cm s - L Using these values of C 1 and C~ and following Hafner [13] Debye temperature O D has been calculated for Cav0Mg30 glass. This result [O D = 347.63 K] has been compared with the value obtained through the expression
PA I - - -P8 ODA2 OD.2
and it deviates by 24%. H e r e PA and Pr~ are the concentration and OD, and ®D, are the Debye temperatures [18] of the pure components. Besides, the isothermal bulk modulus B T of an isotropic solid is given by B T = p ( G 2--4C]2/3), where p is the density of the isotropic solid. For amorphous Ca70Mg30, BT comes out to be 2.437 × 1011 dynes cm 2, which is within 4% of the averaged value for the crystalline metals (BT = 2.32 X 10 j~ dynes cm 2).
4. Conclusion It is found that the new effective pair potential together with the theory of phonons in amorphous solids in a generalized r a n d o m phase approximation gives a fairly accurate description of the vibrational dynamics as well as the thermal and elastic properties of Ca70Mg30 glass.
Acknowledgments Thanks are due to Mr. K. C. Jain and Ms. Neelam Gupta for their help in various ways.
References 1 J. Hafner, in H. Beck and H. J. Giintherodt (eds.), G l a s s y Metals L Topics in Applied Physics, Vol. 64, Springer,
Berlin, 1981, p. 93. 2 J. B. Suck and H. Rudin, in H. Beck and H. J. G6ntherodt (eds.), Glassy Metals" 11, Topics in Applied Physics, Vol. 53, Springer, Berlin, p. 217. 3 L. J, Lewis and N. W. Ashcroft, Phys. Rev., B34 (1986) 8477. 4 M. Rani, A. Pratap and N. S. Saxena, Phys. Status Solidi (b), 149(1988)93.
5 J.Hafner, J. Phys. F, 6 (1976) 1243. 6 J. Hafner. Phys. Rev. A, 16 (1977) 351. 7 Deepika Bhandari, Arun Pratap, N. S. Saxena and M. P. Saksena, Phys. Status Solidi ( b ), 160 (1990) 83. 8 N.W. Ashcroft, Phys. Lett., 23 (1966) 48. 9 J. Hubbard and L. Beeby, J. Phys. C, 2 (1969) 556. 10 J. Hubbard, Proc. R. Soc. London, A243 (1957) 336. L. J. Sham, Proc. R. Soc. London, A283 (1965) 33. 11 J. Hafner and V. Heine, J. Phys. F, 13 (1983) 2479. 12 Arun Pratap, Meeta Rani and N. S. Saxena, Pramana-J. Phys., 30 (1988) 239.
930 13 J. Hafner, Phys. Rev. B, 27(1983) 678. 14 E. Nassif, E Lamparter and S. Steeb, Z. Naturforsch A, 38 (1983) 1206. 15 J. B. Suck, H. Rudin, H. J. Giintherodt, D. Tomanek, H. Beck, C. Morkel and W. Gl~iser, J. Phys. (Paris') Colloq., 41 (1980) C8-175.
16 A. B. Bhatia and R. N. Singhl Phys. Rev. B, 31 (1985) 4751. 17 G. Grimval, Thermophysical Properties of Materials, North-Holland, Amsterdam, 1986. 18 N. W. Ashcroft and D. Stroud, Solid State Physics, 33 Academic Press, New York, 1977, p. 52.