- Email: [email protected]
Computer simulations of vibrational spectra and sound velocity in liquid cesium up to the critical point
Journal of Non-Crystalline Solids 312–314 (2002) 138–142 www.elsevier.com/locate/jnoncrysol
Computer simulations of vibrational spectra and sound velocity in liquid cesium up to the critical point q Boris R. Gelchinski a
a,*
, Alexander A. Mirzoev b, Nikolai P. Smolin
b
Chelyabinsk Center of Ural Branch of the Russian Academy of Sciences, ul. Kommuny, 68, 454000 Chelyabinsk, Russia b Department of General and Theory Physics, Southern Ural State University, 454080 Chelyabinsk, Russia
Abstract The results of ‘phonon’-spectra computer simulations on the basis of the recursion method for liquid cesium for the whole temperature range up to the critical point are presented. The comparison of the experimental data and the calculation results allowed us to conclude that the method gives a precision that is sufficient for the quantitative calculation of the physical properties of melts. We establish that the dispersion curves, xðkÞ, used in analytical theories, in the case of liquid metal has the meaning of a average frequency of a spectrum of collective excitations at the given k value. The application of this method to liquid cesium at various temperatures has shown that at high temperatures there is a qualitative modification of the phonon spectrum that suggests a high localization of the vibration states of liquid metals. The results indicate the presence of a considerable number of highly bound clusters such as dimers, molecular ions or tetrahedrons in liquid metals at high temperatures. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.20.Ja; 61.25.Mv
1. Introduction The dispersion curves of the eigenfrequencies in liquid metals – an analog of the phonon frequency spectra in crystals – can be obtained by neutron diffraction, in which the dynamic structure factor, Sðk; xÞ, [1] is determined. It is known that the magnitude of Sðk; xÞ in liquids makes it possible to determine both the eigenfrequencies of the ‘pho-
q
Supported by Russian Foundation of Basic Researches (RFBR), grants #00-03-32117, #01-03-96467. * Corresponding author. Tel.: +7-351 233 6920; fax: +7-351 233 5672. E-mail address: [email protected] (B.R. Gelchinski).
nons’ and their lifetimes, which are much less than for solid metals. Theoretical methods have also been developed such as in Ref. [2], making it possible to calculate analogs to the solid state phonon spectra, xðkÞ, of topologically disordered systems and their damping. Dispersion curves, xðkÞ, are convenient since that are linear in the limit of small k, which makes it easy to determine the velocities of the longitudinal and the transverse elastic waves in metals. The method described in Ref. [2], originally advanced for the amorphous state, was successfully applied to the theory of liquid metals [3]. Recently, methods of computer simulation, making it possible to avoid many approximations, have played an important role in the theory of
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 6 6 5 - 4
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
liquid metals. In a series of papers [4,5], Hafner has used the recursion method [6] to calculate the phonon spectra of amorphous metals and has showed that the method gives a quantitative fit to the neutron diffraction data. The applicability of this method to describe the dynamic properties of liquid metals is interesting. The purpose of our work is to apply the recursion method in combination with the Reverse Monte Carlo (RMC) method to calculate the vibration properties of melts using liquid cesium as an example. This will allow us to evaluate the applicability of this method to calculate the properties of liquid metals and to explore the atomic dynamics of the metallic melt over a wide range of temperatures. 2. Description of calculations The list of atomic coordinates of the structural models for liquid cesium (2000 particles) over a wide range of temperatures was obtained by constructing structural models using the RMC method. As input data for the RMC model, experimental values of the structure factors of liquid cesium were used for temperatures of 323, 1073, 1673 K [7]. The normal vibrational modes of condensed matter in the harmonic approximation are determined through the eigenvalues, xn , and the eigenvectors, ~ xn ðiÞ, of a dynamic matrix in the coordinate space [8]: Dab ðijÞ ¼ ðMi Mj Þ
1=2
" o2 1X ~ V ð Rk ouia oujb 2 k;m #
Rm ~ um Þ ; þ~ uk ~
account in the Ichimaru–Utsumi approach [10]. As is common, while calculating the power constants matrix, only nearest neighbours interactions were considered. It is possible to show that the local density of states (LDS) for the phonon state spectra with a polarization along the a axis in the neighbourhood of the ith atom is determined by the correlation presented in functional form, using Dirac labelling 2x 1 W ; Im Wj ð2Þ gW ðxÞ ¼ b þ id p x2 D where the state vector, jWi, designates local oscillations of atom i along the a axis. The cross-section for coherent scattering is proportional to the dynamic structure factor Sðk; xÞ [1]: 1=2 d2 r E 2 ¼b Sðk; xÞ; ð3Þ dX dx E0 where E0 and E are the energy of the neutron before and after diffraction, and b is the scattering length of the nuclear potential. In the one-phonon approximation the dynamic structure factor is described by the expression [1] Sðk; xÞ ¼
hp nðxÞ þ 1 2 2W ðkÞ k e f~e¼~k ð~ k ; xÞ; N 2x
~~
where the coefficients i, j enumerate the atomic nodes, a, b ¼ x, y, z, uia is the ath vector component of shift of the ith atom having mass Mi and equilibrium position ~ Ri , V ðrÞ is the interatomic potential. The interatomic potential was calculated in the framework of the pseudopotential method, using the model of non-local potentials [9]. The multi-electron effects of exchange and correlation for the screened pseudopotential were taken into
ð4Þ
where the spectral function as a function of k is defined as 2x X X i~k~Ri k ; xÞ ¼ ea e ImGijab f~e ð~ p ij ab ðx2 þ idÞeb eikRj ;
ð1Þ
139
ð5Þ
This function describes the collective excitations for the wave vector, ~ k , and polarization vector, ~ e. It is possible to represent the spectral function, as a diagonal matrix element from the Green’s function, 2x 1 f~e ð~ ImhW~e jðx2 D þ idÞ jW~e i; k ; xÞ ¼ p _
ð6Þ
where, however, the state vector, jW~e i, now describes the wave shift from the equilibrium state with polarization ~ e: X jW~e i ¼ N 1=2 ea expði~ k ~ Ri Þ; ð7Þ ia
140
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
The diagonal matrix elements of the Green’s function (2) and (6) for the topologically disordered system can be calculated using the recursion method of Haydock [6] in the form of the continued-fraction expansion. Terminating the continued-fraction expansion after L sets of recursion coefficients, which can calculated precisely for the finite size cluster, gives a discrete spectrum of L sharp spectral lines. Any method of analytic continuation of continued fraction for L ! 1 results in replacing the discrete spectrum by a smooth one. In the present work we used the cutoff procedure described in [11], ensuring the greatest smoothing of the spectrum. Normally we calculate the continued-fraction coefficient up to L ¼ 10, which is sufficient at a cluster size of 2000 atoms, but to check the influence of the recursion depth, calculations of up to L ¼ 22 were also made.
3. Results The results of the spectral function, f ðk; xÞ, calculations for liquid cesium at 323 K for longitudinal collective excitations with a wave vector of arbitrary direction in ~ k -space, but a fixed module equal to k, are represented in Fig. 1. The results of the calculation show that the spectral function peak becomes diffuse with increasing of k values it loses its Lorentzian form. The and at k > 0:4 A dispersion law based on xmax becomes indefinite.
Fig. 1. The spectral function, f ðk; xÞ, of the longitudinal collective excitations in liquid Cs at 323 K.
For this reason, we assume a dispersion law for phonons in liquid metals with the dependence R1 xf ðk; xÞ dx xðkÞ ¼ R0 1 ; ð8Þ f ðk; xÞ dx 0 where xðkÞ is an average frequency of the spectrum at the given k. Using dispersion curves for longitudinal oscillations enables us to compare the behavior of the calculated parameters of the spectrum, xmax , and Dx, the dispersion relation, xðkÞ, defined in (8), with the neutron diffraction data [12] (see Fig. 2). It is necessary to note that the same input data about the atomic structure and interionic potential were used in both computational methods for the dispersion relations. From Fig. 2 it is apparent that the dispersion curve, xðkÞ, found using method [2] is close to xmax ðkÞ only at small k, but practically coincides with the average frequency, entered by us, xðkÞ, over the whole calculated range. As it is well known, in a hydrodynamic limit, as k ! 0 xðkÞ ¼ cs k; where cs is the adiabatic sound velocity. The magnitude of the sound velocity should determine the slope of the dispersion curve at low k.
Fig. 2. Dispersion of the longitudinal density fluctuations in liquid cesium at T ¼ 323 K: (1) points gives the frequencies, xmax , for peaks of the spectral function, f ðk; xÞ, the hyphens show the half-width of the spectrum DðxÞ; (2) xðkÞ; (3) xðkÞ – our calculated dispersion relations by method [2]; (4) experimental data [12].
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
141
which have been carried out from (2), are represented in Fig. 4.
4. Discussion
Fig. 3. Dispersion of the longitudinal density fluctuations for liquid cesium and the temperature dependencies of the sound velocity.
Fig. 4. The local density of collective excitations, gðxÞ, in liquid cesium at various temperatures.
The vibration spectra, sound velocity and the dynamic structure factors of liquid cesium were calculated in the temperature range from 323 K up to 1673 K. The calculated temperature dependence of the longitudinal density fluctuations are given in Fig. 3. The results of the calculation of the density of local oscillatory states, gðxÞ, of liquid cesium,
It is easy to see from Fig. 1 that at small k the spectral function has the form of a sharp peak centered close to the definite frequency xmax . This spectral function form makes it possible to discuss the existence of collective excitations in liquid Cs with frequency xmax , which, however, has a finite lifetime proportional to a reciprocal value of the half-width Dx of the peak in question. The existence of a sufficiently well-defined frequency phonon with the present wave vector module makes it possible to use the concept of the excitation dispersion law applicable to liquid metals, defining this as the dependence of xmax ðkÞ. It can be seen in Fig. 3 that at higher temperatures (1673 K) the dispersion law qualitatively changes. This could signal a qualitative reorganization of the atomic structure of the melt. A comparison of the calculated and experimental results for the temperature dependence of the sound velocity is given in the inset frame in Fig. 3. It is apparent that the results of the calculation coincide well with the experimental data [13] at temperatures near the melting point, and that they coincide less and less as the temperature is increased. Close to the melting point, gðxÞ has a continuous spectrum (see Fig. 4). On increasing the temperature to 1073 K the spectrum extends and detaches from the high-frequency peak, corresponding to highly localized vibrations. At 1673 K the area of the continuous spectrum, gðxÞ, moves to the low frequency range, and the density of states is greatly reduced in the rest of the spectrum, which indicates a spatial localization of the oscillations that display themselves as sharp high-frequency peaks. The occurrence of such a quantity of localized states definitely indicates a modification of the atomic structure, related to the occurrence of many small, but strongly bound, clusters (dimers and/or tetrahedrons). For example, in [14] the interpretation of measurements of the electroresistivity in liquid cesium was based on the conjecture of dimers Cs2 and molecular ions Csþ 2 at
142
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
high temperatures. The origin of similar structural formations, such as clusters with strong bonds, should result in the occurrence of features in the range of high frequencies in a spectrum of local excitations in the melt.
Acknowledgements The authors would like to thank Professor R. Winter (Dortmund) for the experimental diffraction data. B.R.G. is also thankful to Professors W. van der Lugt (Groningen) and F. Hensel (Marburg) for their help and hospitality.
5. Conclusion The results of computer simulation and recursion method calculations of frequency spectra for liquid metals using the example of liquid cesium are presented. The comparison of these results with those from data calculated using method [3], and also with the experimental data allowed us to conclude that, despite certain errors connected with the finite cluster size, the method ensures precision sufficient for the quantitative calculation of the physical properties of melts. It was possible to establish that the dispersion law, xðkÞ, used in analytical theories, in the case of liquid metal has the meaning of the k-dependence of the average frequency of the collective excitations spectrum. The application of this method to liquid cesium at various temperatures has shown that at high temperatures there is a qualitative modification of the phonon spectrum, which is evidence for a high localization of the vibration states of liquid metals. The obtained results allow us to conclude that a considerable number of highly bound clusters such as dimers, molecular ions or tetrahedrons form in liquid metals at rather high temperatures.
References [1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart and Winston, New York, 1976. [2] S. Takeno, M. Goda, Progr. Theor. Phys. 45 (1971) 331. [3] A.A. Yuriev, B.R. Gelchinski, V.F. Ukhov, Metallophysica 4 (1982) 108 (in Russian). [4] J. Hafner, Phys. Rev. 27 (1983) 678. [5] J. Hafner, M. Krajci, J. Phys.: Condens. Matter 6 (1994) 4631. [6] R. Haydock, Physics 35 (1980) 215. [7] R. Winter, F. Hensel, T. Bodensteiner, W. Glaser, Ber. Bunsenges. Phys. Chem. 91 (1987) 1327. [8] A. Maradudin, E. Montrol, G. Weiss, I. Ipatova, The Dynamic Theory of a Crystalline Lattice in Harmonic Approach, 2nd Ed., Academic, New York, 1971. [9] B.R. Gelchinski, A.A. Yuriev, N.A. Vatolin, V.F. Ukhov, Doklady AN USSR 261 (1981) 663 (in Russian). [10] S. Ichimaru, K. Utsumi, Phys. Rev. B 24 (1982) 7385. [11] C.M. Nex, Comput. Phys. Commun. 34 (1984) 101. [12] Chr. Morkel, T. Bodensteiner, J. Phys.: Condens. Matter 2 (1990) SA251. [13] P.I. Bystrov, D.N. Kagan, G.A. Krechetova, E.E. Shpilrain, Liquid Metal Coolants for Heat Pipers and Power Plants, Hemisphere, 1990. [14] R. Redmer, H. Reinholz, G. Ropke, R. Winter, F. Noll, F. Hensel, J. Phys.: Condens. Matter. 4 (1992) 1659.
Computer simulations of vibrational spectra and sound velocity in liquid cesium up to the critical point q Boris R. Gelchinski a
a,*
, Alexander A. Mirzoev b, Nikolai P. Smolin
b
Chelyabinsk Center of Ural Branch of the Russian Academy of Sciences, ul. Kommuny, 68, 454000 Chelyabinsk, Russia b Department of General and Theory Physics, Southern Ural State University, 454080 Chelyabinsk, Russia
Abstract The results of ‘phonon’-spectra computer simulations on the basis of the recursion method for liquid cesium for the whole temperature range up to the critical point are presented. The comparison of the experimental data and the calculation results allowed us to conclude that the method gives a precision that is sufficient for the quantitative calculation of the physical properties of melts. We establish that the dispersion curves, xðkÞ, used in analytical theories, in the case of liquid metal has the meaning of a average frequency of a spectrum of collective excitations at the given k value. The application of this method to liquid cesium at various temperatures has shown that at high temperatures there is a qualitative modification of the phonon spectrum that suggests a high localization of the vibration states of liquid metals. The results indicate the presence of a considerable number of highly bound clusters such as dimers, molecular ions or tetrahedrons in liquid metals at high temperatures. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.20.Ja; 61.25.Mv
1. Introduction The dispersion curves of the eigenfrequencies in liquid metals – an analog of the phonon frequency spectra in crystals – can be obtained by neutron diffraction, in which the dynamic structure factor, Sðk; xÞ, [1] is determined. It is known that the magnitude of Sðk; xÞ in liquids makes it possible to determine both the eigenfrequencies of the ‘pho-
q
Supported by Russian Foundation of Basic Researches (RFBR), grants #00-03-32117, #01-03-96467. * Corresponding author. Tel.: +7-351 233 6920; fax: +7-351 233 5672. E-mail address: [email protected] (B.R. Gelchinski).
nons’ and their lifetimes, which are much less than for solid metals. Theoretical methods have also been developed such as in Ref. [2], making it possible to calculate analogs to the solid state phonon spectra, xðkÞ, of topologically disordered systems and their damping. Dispersion curves, xðkÞ, are convenient since that are linear in the limit of small k, which makes it easy to determine the velocities of the longitudinal and the transverse elastic waves in metals. The method described in Ref. [2], originally advanced for the amorphous state, was successfully applied to the theory of liquid metals [3]. Recently, methods of computer simulation, making it possible to avoid many approximations, have played an important role in the theory of
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 6 6 5 - 4
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
liquid metals. In a series of papers [4,5], Hafner has used the recursion method [6] to calculate the phonon spectra of amorphous metals and has showed that the method gives a quantitative fit to the neutron diffraction data. The applicability of this method to describe the dynamic properties of liquid metals is interesting. The purpose of our work is to apply the recursion method in combination with the Reverse Monte Carlo (RMC) method to calculate the vibration properties of melts using liquid cesium as an example. This will allow us to evaluate the applicability of this method to calculate the properties of liquid metals and to explore the atomic dynamics of the metallic melt over a wide range of temperatures. 2. Description of calculations The list of atomic coordinates of the structural models for liquid cesium (2000 particles) over a wide range of temperatures was obtained by constructing structural models using the RMC method. As input data for the RMC model, experimental values of the structure factors of liquid cesium were used for temperatures of 323, 1073, 1673 K [7]. The normal vibrational modes of condensed matter in the harmonic approximation are determined through the eigenvalues, xn , and the eigenvectors, ~ xn ðiÞ, of a dynamic matrix in the coordinate space [8]: Dab ðijÞ ¼ ðMi Mj Þ
1=2
" o2 1X ~ V ð Rk ouia oujb 2 k;m #
Rm ~ um Þ ; þ~ uk ~
account in the Ichimaru–Utsumi approach [10]. As is common, while calculating the power constants matrix, only nearest neighbours interactions were considered. It is possible to show that the local density of states (LDS) for the phonon state spectra with a polarization along the a axis in the neighbourhood of the ith atom is determined by the correlation presented in functional form, using Dirac labelling 2x 1 W ; Im Wj ð2Þ gW ðxÞ ¼ b þ id p x2 D where the state vector, jWi, designates local oscillations of atom i along the a axis. The cross-section for coherent scattering is proportional to the dynamic structure factor Sðk; xÞ [1]: 1=2 d2 r E 2 ¼b Sðk; xÞ; ð3Þ dX dx E0 where E0 and E are the energy of the neutron before and after diffraction, and b is the scattering length of the nuclear potential. In the one-phonon approximation the dynamic structure factor is described by the expression [1] Sðk; xÞ ¼
hp nðxÞ þ 1 2 2W ðkÞ k e f~e¼~k ð~ k ; xÞ; N 2x
~~
where the coefficients i, j enumerate the atomic nodes, a, b ¼ x, y, z, uia is the ath vector component of shift of the ith atom having mass Mi and equilibrium position ~ Ri , V ðrÞ is the interatomic potential. The interatomic potential was calculated in the framework of the pseudopotential method, using the model of non-local potentials [9]. The multi-electron effects of exchange and correlation for the screened pseudopotential were taken into
ð4Þ
where the spectral function as a function of k is defined as 2x X X i~k~Ri k ; xÞ ¼ ea e ImGijab f~e ð~ p ij ab ðx2 þ idÞeb eikRj ;
ð1Þ
139
ð5Þ
This function describes the collective excitations for the wave vector, ~ k , and polarization vector, ~ e. It is possible to represent the spectral function, as a diagonal matrix element from the Green’s function, 2x 1 f~e ð~ ImhW~e jðx2 D þ idÞ jW~e i; k ; xÞ ¼ p _
ð6Þ
where, however, the state vector, jW~e i, now describes the wave shift from the equilibrium state with polarization ~ e: X jW~e i ¼ N 1=2 ea expði~ k ~ Ri Þ; ð7Þ ia
140
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
The diagonal matrix elements of the Green’s function (2) and (6) for the topologically disordered system can be calculated using the recursion method of Haydock [6] in the form of the continued-fraction expansion. Terminating the continued-fraction expansion after L sets of recursion coefficients, which can calculated precisely for the finite size cluster, gives a discrete spectrum of L sharp spectral lines. Any method of analytic continuation of continued fraction for L ! 1 results in replacing the discrete spectrum by a smooth one. In the present work we used the cutoff procedure described in [11], ensuring the greatest smoothing of the spectrum. Normally we calculate the continued-fraction coefficient up to L ¼ 10, which is sufficient at a cluster size of 2000 atoms, but to check the influence of the recursion depth, calculations of up to L ¼ 22 were also made.
3. Results The results of the spectral function, f ðk; xÞ, calculations for liquid cesium at 323 K for longitudinal collective excitations with a wave vector of arbitrary direction in ~ k -space, but a fixed module equal to k, are represented in Fig. 1. The results of the calculation show that the spectral function peak becomes diffuse with increasing of k values it loses its Lorentzian form. The and at k > 0:4 A dispersion law based on xmax becomes indefinite.
Fig. 1. The spectral function, f ðk; xÞ, of the longitudinal collective excitations in liquid Cs at 323 K.
For this reason, we assume a dispersion law for phonons in liquid metals with the dependence R1 xf ðk; xÞ dx xðkÞ ¼ R0 1 ; ð8Þ f ðk; xÞ dx 0 where xðkÞ is an average frequency of the spectrum at the given k. Using dispersion curves for longitudinal oscillations enables us to compare the behavior of the calculated parameters of the spectrum, xmax , and Dx, the dispersion relation, xðkÞ, defined in (8), with the neutron diffraction data [12] (see Fig. 2). It is necessary to note that the same input data about the atomic structure and interionic potential were used in both computational methods for the dispersion relations. From Fig. 2 it is apparent that the dispersion curve, xðkÞ, found using method [2] is close to xmax ðkÞ only at small k, but practically coincides with the average frequency, entered by us, xðkÞ, over the whole calculated range. As it is well known, in a hydrodynamic limit, as k ! 0 xðkÞ ¼ cs k; where cs is the adiabatic sound velocity. The magnitude of the sound velocity should determine the slope of the dispersion curve at low k.
Fig. 2. Dispersion of the longitudinal density fluctuations in liquid cesium at T ¼ 323 K: (1) points gives the frequencies, xmax , for peaks of the spectral function, f ðk; xÞ, the hyphens show the half-width of the spectrum DðxÞ; (2) xðkÞ; (3) xðkÞ – our calculated dispersion relations by method [2]; (4) experimental data [12].
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
141
which have been carried out from (2), are represented in Fig. 4.
4. Discussion
Fig. 3. Dispersion of the longitudinal density fluctuations for liquid cesium and the temperature dependencies of the sound velocity.
Fig. 4. The local density of collective excitations, gðxÞ, in liquid cesium at various temperatures.
The vibration spectra, sound velocity and the dynamic structure factors of liquid cesium were calculated in the temperature range from 323 K up to 1673 K. The calculated temperature dependence of the longitudinal density fluctuations are given in Fig. 3. The results of the calculation of the density of local oscillatory states, gðxÞ, of liquid cesium,
It is easy to see from Fig. 1 that at small k the spectral function has the form of a sharp peak centered close to the definite frequency xmax . This spectral function form makes it possible to discuss the existence of collective excitations in liquid Cs with frequency xmax , which, however, has a finite lifetime proportional to a reciprocal value of the half-width Dx of the peak in question. The existence of a sufficiently well-defined frequency phonon with the present wave vector module makes it possible to use the concept of the excitation dispersion law applicable to liquid metals, defining this as the dependence of xmax ðkÞ. It can be seen in Fig. 3 that at higher temperatures (1673 K) the dispersion law qualitatively changes. This could signal a qualitative reorganization of the atomic structure of the melt. A comparison of the calculated and experimental results for the temperature dependence of the sound velocity is given in the inset frame in Fig. 3. It is apparent that the results of the calculation coincide well with the experimental data [13] at temperatures near the melting point, and that they coincide less and less as the temperature is increased. Close to the melting point, gðxÞ has a continuous spectrum (see Fig. 4). On increasing the temperature to 1073 K the spectrum extends and detaches from the high-frequency peak, corresponding to highly localized vibrations. At 1673 K the area of the continuous spectrum, gðxÞ, moves to the low frequency range, and the density of states is greatly reduced in the rest of the spectrum, which indicates a spatial localization of the oscillations that display themselves as sharp high-frequency peaks. The occurrence of such a quantity of localized states definitely indicates a modification of the atomic structure, related to the occurrence of many small, but strongly bound, clusters (dimers and/or tetrahedrons). For example, in [14] the interpretation of measurements of the electroresistivity in liquid cesium was based on the conjecture of dimers Cs2 and molecular ions Csþ 2 at
142
B.R. Gelchinski et al. / Journal of Non-Crystalline Solids 312–314 (2002) 138–142
high temperatures. The origin of similar structural formations, such as clusters with strong bonds, should result in the occurrence of features in the range of high frequencies in a spectrum of local excitations in the melt.
Acknowledgements The authors would like to thank Professor R. Winter (Dortmund) for the experimental diffraction data. B.R.G. is also thankful to Professors W. van der Lugt (Groningen) and F. Hensel (Marburg) for their help and hospitality.
5. Conclusion The results of computer simulation and recursion method calculations of frequency spectra for liquid metals using the example of liquid cesium are presented. The comparison of these results with those from data calculated using method [3], and also with the experimental data allowed us to conclude that, despite certain errors connected with the finite cluster size, the method ensures precision sufficient for the quantitative calculation of the physical properties of melts. It was possible to establish that the dispersion law, xðkÞ, used in analytical theories, in the case of liquid metal has the meaning of the k-dependence of the average frequency of the collective excitations spectrum. The application of this method to liquid cesium at various temperatures has shown that at high temperatures there is a qualitative modification of the phonon spectrum, which is evidence for a high localization of the vibration states of liquid metals. The obtained results allow us to conclude that a considerable number of highly bound clusters such as dimers, molecular ions or tetrahedrons form in liquid metals at rather high temperatures.
References [1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart and Winston, New York, 1976. [2] S. Takeno, M. Goda, Progr. Theor. Phys. 45 (1971) 331. [3] A.A. Yuriev, B.R. Gelchinski, V.F. Ukhov, Metallophysica 4 (1982) 108 (in Russian). [4] J. Hafner, Phys. Rev. 27 (1983) 678. [5] J. Hafner, M. Krajci, J. Phys.: Condens. Matter 6 (1994) 4631. [6] R. Haydock, Physics 35 (1980) 215. [7] R. Winter, F. Hensel, T. Bodensteiner, W. Glaser, Ber. Bunsenges. Phys. Chem. 91 (1987) 1327. [8] A. Maradudin, E. Montrol, G. Weiss, I. Ipatova, The Dynamic Theory of a Crystalline Lattice in Harmonic Approach, 2nd Ed., Academic, New York, 1971. [9] B.R. Gelchinski, A.A. Yuriev, N.A. Vatolin, V.F. Ukhov, Doklady AN USSR 261 (1981) 663 (in Russian). [10] S. Ichimaru, K. Utsumi, Phys. Rev. B 24 (1982) 7385. [11] C.M. Nex, Comput. Phys. Commun. 34 (1984) 101. [12] Chr. Morkel, T. Bodensteiner, J. Phys.: Condens. Matter 2 (1990) SA251. [13] P.I. Bystrov, D.N. Kagan, G.A. Krechetova, E.E. Shpilrain, Liquid Metal Coolants for Heat Pipers and Power Plants, Hemisphere, 1990. [14] R. Redmer, H. Reinholz, G. Ropke, R. Winter, F. Noll, F. Hensel, J. Phys.: Condens. Matter. 4 (1992) 1659.