- Email: [email protected]
Isothermal compressibility and sound velocity of liquid rare earth metals
Journal of the Less-Common
Metals,
136 (1987)
ISOTHERMAL COMPRESSIBILITY LIQUID RARE EARTH METALS I. YOKOYAMA Department
25 - 29
25
AND SOUND VELOCITY
OF
and S. NAITO
of Mathematics
and Physics,
The National
Defense Academy,
Yokosuka
239
fJw4 Y. WASEDA The Research Institute of Mineral Dressing and Metallurgy Tohoku University, Sendai 980 (Japan)
(SENKEN),
(Received March 16,1987)
Summary The simple one-component plasma (OCP) model has been applied to the isothermal compressibilities of liquid rare earth metals. The calculated results are in fair agreement with those estimated from the long-wavelength limit of liquid structure factors and the well-known relation between the isothermal compressibility and the surface tension. The velocity of sound has been calculated by the Percus-Yevick (PY) phonon model using recently measured low-angle structure data. The OCP model appears to provide a fair description of the thermodyn~ic properties of liquid rare earth metals.
1. Introduction Recently Itami and Shimoji [l] have shown that the one-component plasma (OCP) model [Z - 41 is capable of describing the thermodynamic properties of liquid 3d tr~sition metals close to their melting points. This approach was further investigated by Itoh et ~2. [ 5 ] and its usefulness for transition metals has been discussed in comparison with the results obtained by the Percus-Yevick (PY) phonon model [6, 73. For liquid rare earth metals, however, only the structural features [S] and the entropies [9] have been studied in terms of the OCP model. The purpose of this paper is twofold: firstly, to calculate the isothermal compressib~ity using the OCP model approach of Itami and Shimoji so as to confirm its utility in predicting the the~odyn~ic properties of the liquid rare earths; secondly, to predict the velocity of sound of these liquid metals, in terms of the PY phonon description, which has not yet been reported. In a previous paper [9] we concentrated on the entropy and the specific heat at constant volume, and this paper may be viewed as an extension of that work. 0022-5088/S7t$3.50
@ Elsevier Sequoia/Printed
in The Netherlands
26 2.
Theory
2.1. The OCP model The Helmholtz free energy of a liquid rare earth metal is now assumed to be given by F=F,+U,(V,T)
(1)
where F, is the Helmholtz free energy of a model reference liquid capable of describing the structuredependent part of the system of interest, and U,( V, T) is a function of volume V and temperature T representing the difference between the free energies of the reference and the real liquid. If the OCP model is taken as the reference liquid, the entropy S and the isothermal compre~ib~ty XT are given as follows (see refs. 1, 5, 9 and 10 for further details): S = S,,, + ASOCP
+ $1
+ &nag
(2)
where
ASOCP NkB
-
St31
=
1 3
n2kBT
(5)
NkB
S
-..?IZ = ln(2J + 1)
(6)
mB
and 13
-Br”4+
144
+- V a2W,,odV, ‘O/N1 N [ a(vo2 T
I
(7)
In these equations N denotes the number of ions, M the mass of an ion, kg the Boltzmann constant, h the Planck constant and l? the plasma parameter which is defined by I’ = (.&)*/Uk,T with 2, e and a being the valency, the electron charge and the Wigner-Seitz radius (a = (3V/47rN)“3) respectively. The constants A, B, C and D are respectively -0.896 434, 3.447 408, -0.555 130 and -2.995 974 [II]. J represents the total angular momentum of the ground state f electrons and N(Er) is the density of states at the Fermi level, the values of which have been estimated by Harder and Young
Iwl.
21
2.2. The PY phonon model We imagine a liquid to possess 3N normal density-fluctuation modes. The work of March and coworkers [12,13] suggests that this is true for liquid metals but not for liquid argon. This is because of the rather different characters of the interatomic forces involved. According to this model, the dispersion relation, to the first order of phonon perturbation, is given by ]14,151
(8) where WEis the independent-phonon frequency, given by Mk,T -u(k)x2 fi2k2
X
fiw,”
x= (9) kBT In the above equations, k is the wave number, A = h/2n and a(k) is the static structure factor. In the subsequent numerical processing of these equations, we use the observed a(k) and the form {a In a(k/ko)/a In T}, proposed in ref. 14. The temperature dependence of the structure factor at constant volume has not been measured yet, but a feasible form applicable to all liquid metals near their melting points is suggested in Fig. 1 of ref. 14. Since the sound velocity is defined as = e”-1
+ Ix 2
+a(k)
‘dw (19) ( dk 1k=O the slope of the phonon dispersion curve at sufficiently small k should give the velocity of sound at the fixed temperature and density. u=
3. Results and discussion The calculated thermodynamic quantities are summarized in Table 1 together with the input data and the experimental values. The experimental structure factors are used directly for the evaluation of the sound velocity. The OCP structure factors proposed by Chaturvedi et al. [19,20] are used for the OCP model, where the optimum values of r have been determined so as to describe the observed structure factor reasonably well. With the values of r thus determined, the entropy S and the isothermal compressibility xT are calculated. Values of r slightly different from those in ref. 9 were used in some cases, but they gave a better overall description of the observed structure data. Therefore, we have recalculated the entropies and compared them with the observed data. The conclusion stated in ref. 9 for the entropies is confirmed. The effective valences Zr obtained from the plasma parameters are roughly between 1.2 and 1.7, evidently smaller than those in the solid state (three valence electrons per atom are generally accepted for all rare earth metals except europium and ytterbium). Because of the presence
28 TABLE 1 Entropy S (in units of iVkB), isothermal compressibility XT (lo-” m* N-l), adiabatic compressibility xs (lo-” m* N-l) and sound velocity u (m 8-l) of liquid rare earth metals. The Zp is the effective valence calculated from the plasma parameter r with which the observed structure factor can be moderately well described by the OCP structure factor.
La Ce Pr Nd Eu Gd Tb DY Ho Er Yb Lu
T(K)
r
-+
x~cacp Gi.% sexpta
x$alc
1243 1143 1223 1323 1103 1603 1653 1703 1753 1793 1123 1953
100 95 90 110 100 110 110 110 130 130 110 130
1.25 1.15 1.16 1.33 1.25 1.48 1.50 1.49 1.65 1.67 1.28 1.73
13.0 14.7 15.1 15.2 15.2 16.0 16.6
13.0 14.4 15.5 15.7 15.6 15.6 16.8
13.0
13.0
4.37 4.11 4.16 3.75 7.20 3.25 3.10 2.68 2.49 2.23 5.34 1.95
4.64 4.80 4.82 3.64 6.97 3.29 3.11 2.79 2.31 2.26 5.41 2.01
b
Xc+xptc 4.29 4.64 5.23 3.41 6.46 3.07 3.40 2.97 1.92 2.28 4.48 1.88
talc d
%alc
x8
2080 1910 1800 2200 1860 2240 2120 2130 2560 2450 1920 2380
3.88 4.11 4.67 2.99 6.27 2.88 3.07 2.71 1.85 1.99 4.38 1.92
*Taken from Hultgren et al. [ 161. bCalculated from the empirical relation xyy/l= 0.058 where 7 denotes the surface tension and 1the surface thickness [ 17 1. CEvaluated from the long-wavelength value of diffraction data [ 181. dCalculated from xs = l/pu* where p is the mass density.
of d or f states, the physical meaning of Zr is not obvious and we rather regard it as an adjustable parameter to simulate a liquid metal by the OCP model. For the isothermal compressibility it will be seen that the values of X$LP are in fair agreement with the values of Xcxpt and XF”, although X$+&r, seems to be a little larger on the whole than XFxpt. The factor
in eqn. (7) may be responsible for this. To evaluate that term the derivatives of N(E,) and J with respect to the density are needed, but they are completely neglected in this work because, to the best knowledge of the authors, there is no information about them. For the velocity of sound, there is no experimental data available for comparison. However, we believe the estimated values could be reliable in view of the calculations reported in previous work [ 14,151. The adiabatic compressibilities X8 are easily obtained from the calculated velocities of sound and these have been given in the final column of Table 1, enabling the readers to deduce the specific heat ratio.
29
4. Conclusion The simple OCP model appears to describe well the entropies and the isothermal compressibilities of liquid rare earth metals. For a fully quantitative discussion, however, information on the density derivatives of N(Er) and J seems to be essential. Also, accurate experimental values of the sound velocities are highly desirable to permit the values calculated on the basis of the PY phonon model to be checked.
Acknowledgment I.Y. and Y .W. are grateful for partial support from the Kawasaki Steel Corporation, Technical Research Division.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20
T. Itami and M. Shimoji, J. Phys. F, 14 (1984) L15. M. Baus and J.-P. Hansen, Phys. Rep., 59 (1980) 1. M. K. Mon. R. Gann and D. Stroud, Phys. Rev. A, 24 (1981) 2145. M. Ross, H. E. Dewitt and W. B. Hubbard,Phys. Rev. A, 24 (1981) 1016. H. Itoh, I. Yokoyama and Y. Waseda, J. Phys. F, 16 (1986) L113. P. Gray, I. Yokoyama and W. H. Young, J. Phys. F, 10 (1980) 197. I. Ohkoshi, I. Yokoyama, Y. Waseda and W. H. Young, J. Phys. F, 11 (1981) 531. S. N. Khanna and F. Cyrot-Lackmann, J. Phys. (Paris), Lett., 40 (1979) L45. I. Yokoyama, S. Naito and Y. Waseda, Proc. 6th Int. Conf. on Liquid and Amorphous Metals. 1987,Z. Phys. Chem., in the press. J. M. Harder and W. H. Young,Phys. Lett. A, 61 (1977) 468. S. Galam and J.-P. Hansen, Phys. Rev. A, 14 (1976) 816. P. Bratby, T. Gaskell and N. H. March, Phys. Chem. Liq., 2 (1970) 53. N. H. March and M. P. Tosi, Atomic Dynamics in Liquids, MacmiIIan, London, 1976. I. Yokoyama, I. Ohkoshi and W. H. Young, Phys. Chem. Lip., 11 (1981) 179. I. Yokoyama, I. Ohkoshi, Y. Waseda and W. H. Young, Phys. Chem. Liq., 11 (1982) 277. R, Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. KeiIey and D. Wagman, Selected Values of the Thermodynamic Properties of the Elements, American Society for Metals, MetaIs Park, OH, 1973. Y. Waseda and K. T. Jacob, Phys. Status Solidi A, 68 (1981) K117. Y. Waseda and S. Ueno, Sci. Rep. Res. Inst., Tohoku Univ., Ser. A, 34 (1987) 1. D. K. Chaturvedi, G. Senatore and M. P. Tosi, Left. Nuovo Cimento, 30 (1981) 47. D. K. Chaturvedi, G. Senatore and M. P. Tosi, Nuovo Cimento B, 62 (1981) 375.
Metals,
136 (1987)
ISOTHERMAL COMPRESSIBILITY LIQUID RARE EARTH METALS I. YOKOYAMA Department
25 - 29
25
AND SOUND VELOCITY
OF
and S. NAITO
of Mathematics
and Physics,
The National
Defense Academy,
Yokosuka
239
fJw4 Y. WASEDA The Research Institute of Mineral Dressing and Metallurgy Tohoku University, Sendai 980 (Japan)
(SENKEN),
(Received March 16,1987)
Summary The simple one-component plasma (OCP) model has been applied to the isothermal compressibilities of liquid rare earth metals. The calculated results are in fair agreement with those estimated from the long-wavelength limit of liquid structure factors and the well-known relation between the isothermal compressibility and the surface tension. The velocity of sound has been calculated by the Percus-Yevick (PY) phonon model using recently measured low-angle structure data. The OCP model appears to provide a fair description of the thermodyn~ic properties of liquid rare earth metals.
1. Introduction Recently Itami and Shimoji [l] have shown that the one-component plasma (OCP) model [Z - 41 is capable of describing the thermodynamic properties of liquid 3d tr~sition metals close to their melting points. This approach was further investigated by Itoh et ~2. [ 5 ] and its usefulness for transition metals has been discussed in comparison with the results obtained by the Percus-Yevick (PY) phonon model [6, 73. For liquid rare earth metals, however, only the structural features [S] and the entropies [9] have been studied in terms of the OCP model. The purpose of this paper is twofold: firstly, to calculate the isothermal compressib~ity using the OCP model approach of Itami and Shimoji so as to confirm its utility in predicting the the~odyn~ic properties of the liquid rare earths; secondly, to predict the velocity of sound of these liquid metals, in terms of the PY phonon description, which has not yet been reported. In a previous paper [9] we concentrated on the entropy and the specific heat at constant volume, and this paper may be viewed as an extension of that work. 0022-5088/S7t$3.50
@ Elsevier Sequoia/Printed
in The Netherlands
26 2.
Theory
2.1. The OCP model The Helmholtz free energy of a liquid rare earth metal is now assumed to be given by F=F,+U,(V,T)
(1)
where F, is the Helmholtz free energy of a model reference liquid capable of describing the structuredependent part of the system of interest, and U,( V, T) is a function of volume V and temperature T representing the difference between the free energies of the reference and the real liquid. If the OCP model is taken as the reference liquid, the entropy S and the isothermal compre~ib~ty XT are given as follows (see refs. 1, 5, 9 and 10 for further details): S = S,,, + ASOCP
+ $1
+ &nag
(2)
where
ASOCP NkB
-
St31
=
1 3
n2kBT
(5)
NkB
S
-..?IZ = ln(2J + 1)
(6)
mB
and 13
-Br”4+
144
+- V a2W,,odV, ‘O/N1 N [ a(vo2 T
I
(7)
In these equations N denotes the number of ions, M the mass of an ion, kg the Boltzmann constant, h the Planck constant and l? the plasma parameter which is defined by I’ = (.&)*/Uk,T with 2, e and a being the valency, the electron charge and the Wigner-Seitz radius (a = (3V/47rN)“3) respectively. The constants A, B, C and D are respectively -0.896 434, 3.447 408, -0.555 130 and -2.995 974 [II]. J represents the total angular momentum of the ground state f electrons and N(Er) is the density of states at the Fermi level, the values of which have been estimated by Harder and Young
Iwl.
21
2.2. The PY phonon model We imagine a liquid to possess 3N normal density-fluctuation modes. The work of March and coworkers [12,13] suggests that this is true for liquid metals but not for liquid argon. This is because of the rather different characters of the interatomic forces involved. According to this model, the dispersion relation, to the first order of phonon perturbation, is given by ]14,151
(8) where WEis the independent-phonon frequency, given by Mk,T -u(k)x2 fi2k2
X
fiw,”
x= (9) kBT In the above equations, k is the wave number, A = h/2n and a(k) is the static structure factor. In the subsequent numerical processing of these equations, we use the observed a(k) and the form {a In a(k/ko)/a In T}, proposed in ref. 14. The temperature dependence of the structure factor at constant volume has not been measured yet, but a feasible form applicable to all liquid metals near their melting points is suggested in Fig. 1 of ref. 14. Since the sound velocity is defined as = e”-1
+ Ix 2
+a(k)
‘dw (19) ( dk 1k=O the slope of the phonon dispersion curve at sufficiently small k should give the velocity of sound at the fixed temperature and density. u=
3. Results and discussion The calculated thermodynamic quantities are summarized in Table 1 together with the input data and the experimental values. The experimental structure factors are used directly for the evaluation of the sound velocity. The OCP structure factors proposed by Chaturvedi et al. [19,20] are used for the OCP model, where the optimum values of r have been determined so as to describe the observed structure factor reasonably well. With the values of r thus determined, the entropy S and the isothermal compressibility xT are calculated. Values of r slightly different from those in ref. 9 were used in some cases, but they gave a better overall description of the observed structure data. Therefore, we have recalculated the entropies and compared them with the observed data. The conclusion stated in ref. 9 for the entropies is confirmed. The effective valences Zr obtained from the plasma parameters are roughly between 1.2 and 1.7, evidently smaller than those in the solid state (three valence electrons per atom are generally accepted for all rare earth metals except europium and ytterbium). Because of the presence
28 TABLE 1 Entropy S (in units of iVkB), isothermal compressibility XT (lo-” m* N-l), adiabatic compressibility xs (lo-” m* N-l) and sound velocity u (m 8-l) of liquid rare earth metals. The Zp is the effective valence calculated from the plasma parameter r with which the observed structure factor can be moderately well described by the OCP structure factor.
La Ce Pr Nd Eu Gd Tb DY Ho Er Yb Lu
T(K)
r
-+
x~cacp Gi.% sexpta
x$alc
1243 1143 1223 1323 1103 1603 1653 1703 1753 1793 1123 1953
100 95 90 110 100 110 110 110 130 130 110 130
1.25 1.15 1.16 1.33 1.25 1.48 1.50 1.49 1.65 1.67 1.28 1.73
13.0 14.7 15.1 15.2 15.2 16.0 16.6
13.0 14.4 15.5 15.7 15.6 15.6 16.8
13.0
13.0
4.37 4.11 4.16 3.75 7.20 3.25 3.10 2.68 2.49 2.23 5.34 1.95
4.64 4.80 4.82 3.64 6.97 3.29 3.11 2.79 2.31 2.26 5.41 2.01
b
Xc+xptc 4.29 4.64 5.23 3.41 6.46 3.07 3.40 2.97 1.92 2.28 4.48 1.88
talc d
%alc
x8
2080 1910 1800 2200 1860 2240 2120 2130 2560 2450 1920 2380
3.88 4.11 4.67 2.99 6.27 2.88 3.07 2.71 1.85 1.99 4.38 1.92
*Taken from Hultgren et al. [ 161. bCalculated from the empirical relation xyy/l= 0.058 where 7 denotes the surface tension and 1the surface thickness [ 17 1. CEvaluated from the long-wavelength value of diffraction data [ 181. dCalculated from xs = l/pu* where p is the mass density.
of d or f states, the physical meaning of Zr is not obvious and we rather regard it as an adjustable parameter to simulate a liquid metal by the OCP model. For the isothermal compressibility it will be seen that the values of X$LP are in fair agreement with the values of Xcxpt and XF”, although X$+&r, seems to be a little larger on the whole than XFxpt. The factor
in eqn. (7) may be responsible for this. To evaluate that term the derivatives of N(E,) and J with respect to the density are needed, but they are completely neglected in this work because, to the best knowledge of the authors, there is no information about them. For the velocity of sound, there is no experimental data available for comparison. However, we believe the estimated values could be reliable in view of the calculations reported in previous work [ 14,151. The adiabatic compressibilities X8 are easily obtained from the calculated velocities of sound and these have been given in the final column of Table 1, enabling the readers to deduce the specific heat ratio.
29
4. Conclusion The simple OCP model appears to describe well the entropies and the isothermal compressibilities of liquid rare earth metals. For a fully quantitative discussion, however, information on the density derivatives of N(Er) and J seems to be essential. Also, accurate experimental values of the sound velocities are highly desirable to permit the values calculated on the basis of the PY phonon model to be checked.
Acknowledgment I.Y. and Y .W. are grateful for partial support from the Kawasaki Steel Corporation, Technical Research Division.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20
T. Itami and M. Shimoji, J. Phys. F, 14 (1984) L15. M. Baus and J.-P. Hansen, Phys. Rep., 59 (1980) 1. M. K. Mon. R. Gann and D. Stroud, Phys. Rev. A, 24 (1981) 2145. M. Ross, H. E. Dewitt and W. B. Hubbard,Phys. Rev. A, 24 (1981) 1016. H. Itoh, I. Yokoyama and Y. Waseda, J. Phys. F, 16 (1986) L113. P. Gray, I. Yokoyama and W. H. Young, J. Phys. F, 10 (1980) 197. I. Ohkoshi, I. Yokoyama, Y. Waseda and W. H. Young, J. Phys. F, 11 (1981) 531. S. N. Khanna and F. Cyrot-Lackmann, J. Phys. (Paris), Lett., 40 (1979) L45. I. Yokoyama, S. Naito and Y. Waseda, Proc. 6th Int. Conf. on Liquid and Amorphous Metals. 1987,Z. Phys. Chem., in the press. J. M. Harder and W. H. Young,Phys. Lett. A, 61 (1977) 468. S. Galam and J.-P. Hansen, Phys. Rev. A, 14 (1976) 816. P. Bratby, T. Gaskell and N. H. March, Phys. Chem. Liq., 2 (1970) 53. N. H. March and M. P. Tosi, Atomic Dynamics in Liquids, MacmiIIan, London, 1976. I. Yokoyama, I. Ohkoshi and W. H. Young, Phys. Chem. Lip., 11 (1981) 179. I. Yokoyama, I. Ohkoshi, Y. Waseda and W. H. Young, Phys. Chem. Liq., 11 (1982) 277. R, Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. KeiIey and D. Wagman, Selected Values of the Thermodynamic Properties of the Elements, American Society for Metals, MetaIs Park, OH, 1973. Y. Waseda and K. T. Jacob, Phys. Status Solidi A, 68 (1981) K117. Y. Waseda and S. Ueno, Sci. Rep. Res. Inst., Tohoku Univ., Ser. A, 34 (1987) 1. D. K. Chaturvedi, G. Senatore and M. P. Tosi, Left. Nuovo Cimento, 30 (1981) 47. D. K. Chaturvedi, G. Senatore and M. P. Tosi, Nuovo Cimento B, 62 (1981) 375.