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Dynamical properties of liquid phosphorus studied by ab initio molecular-dynamics simulations
Journal of Non-Crystalline Solids 353 (2007) 3488–3491 www.elsevier.com/locate/jnoncrysol
Dynamical properties of liquid phosphorus studied by ab initio molecular-dynamics simulations Yasuhiro Senda
a,*
, Fuyuki Shimojo b, Kozo Hoshino
c
a
c
Department of Computational Science, Kanazawa University, Kanazawa 920-1192, Japan b Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan Available online 27 July 2007
Abstract We have performed ab initio molecular-dynamics simulations for the low-density and high-density liquid phosphorus to investigate their dynamical properties. We have shown that: (i) The vibration modes within the tetrahedral P4 molecules in the low-density molecular liquid can be seen in the calculated velocity autocorrelation function and its spectrum. (ii) A so-called de Gennes narrowing due to the strong spatial correlation between the P4 molecules and phonon peaks are seen in the calculated dynamic structure factors. (iii) The sound velocities estimated from the dispersion relation obtained from the dynamic structure factors are 2100 m s1 for the molecular liquid and 5300 m s1 for the polymeric liquid, which reflect the differences in the density and in the liquid state. Ó 2007 Published by Elsevier B.V. PACS: 61.20.Ja; 61.20.Lc Keywords: Acoustic properties and phonons; Liquid alloys and liquid metals; Diffraction and scattering measurements; Modeling and simulation; Ab initio; Molecular-dynamics; Structure
1. Introduction It is well known that phosphorus (P) shows a variety of structures for a wide range of pressures and temperatures; a molecular solid, the so-called ‘white’ phosphorus, is composed of tetrahedral P4 clusters, a variety of ‘red’ phosphorus that are usually amorphous, and an orthorhombic structure of ‘black’ phosphorus that consists of puckered double layers of sixfold rings. Katayama et al. [1] reported an X-ray diffraction observation of an abrupt pressure-induced structural change in liquid phosphorus at about 1 GPa as shown in Fig. 1. The transformation was expected to be a first-order liquid–
* Corresponding author. Present address: Graduate School of Science and Engineering, Yamaguchi University, Ube 755-8611, Japan. Tel.: +81 836 85 9812; fax: +81 836 85 9800. E-mail address: [email protected] (Y. Senda).
0022-3093/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.jnoncrysol.2007.05.102
liquid structural phase transition, accompanied by a discontinuous change of density of the liquid phosphorus. It was clearly shown by ab initio molecular-dynamics (MD) simulations [2,3] that the structural phase transition found by Katayama et al. [1] corresponds to the structural change from the molecular liquid composed of stable tetrahedral P4 molecules to the polymeric liquid with complex network structure. The calculated structure factors are in good agreement with those obtained by the X-ray diffraction experiments as shown in Fig. 1 and the characteristic features of the observed S(k) of liquid phosphorus for lower and higher pressures (densities) are well reproduced by ab initio MD simulations. It was also found that the structural change gives rise to the nonmetal–metal transition [3]. It is expected from these results for liquid phosphorus that the dynamical properties of the low-density and highdensity liquid phosphorus are also quite different. Though recent progress in inelastic X-ray scattering experiments
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supercell is used and periodic boundary conditions are imposed. The total number of atoms in the supercell is taken to be 100. As an initial configuration, P4 tetrahedra are arranged on the simple-cubic lattice. A constant-temperature MD simulation is carried out using the Nose´Hoover thermostat technique at the temperature of 1350 K. The observed densities [1,13] are used in our calculation. Since long time simulation is required for the determination of reliable dynamical quantities, we carry out our simulation for 24 ps with a time step of 2.4 fs after the thermal equilibrium state is achieved. 3. Results
Fig. 1. The S(k) for the liquid phosphorus at low-density (circles) and at high-density (squares). The full-line shows the experimental results at the corresponding pressures.
enables us to observe the dynamical properties of some liquids at high temperatures [4–7], little is known so far about the dynamical properties of the liquid phosphorus. In the present study, we carry out ab initio MD simulations for low-density and high-density liquid phosphorus to investigate the dynamical properties of them. We obtain the velocity autocorrelation functions and the dynamic structure factors and estimate the velocity of sound for these liquid phosphorus to discuss the dynamical properties of them comparatively.
The calculated velocity autocorrelation function Z(t), ^ and its spectrum ZðxÞ are shown in Fig. 2(a) and (b), respectively. As shown in Fig. 2(a), Z(t) of the low-density molecular liquid show an oscillating behaviour, which is related to the intramolecular vibration. It is shown in Fig. 2(b) that the low-density liquid has three peaks at x = 42, 53, ^ 70 meV in the ZðxÞ besides the diffusive mode in low-x region, while the high-density liquid has a shoulder at x = 40 meV. The frequencies of the P4 molecule in gas phase, x = 45.9, 57.9, 76.0 meV, observed by the Raman spectroscopy [14,15] are also indicated by arrows in Fig. 2(b). We have obtained by our simulation the intermediate scattering function F(k, t) defined by F ðk; tÞ ¼
1 hqðk; tÞqðk; 0Þi N
2. Method of calculation To study the structure and the electronic states of liquid phosphorus, we carry out an ab initio MD simulation, which is based on the density functional theory in the local-density approximation, the pseudopotential theory and the adiabatic approximation. In our calculation, we minimize the Kohn–Sham energy functional for a given ionic configuration at each time step by the conjugate-gradient method [8] and calculate the electron density and the forces acting on ions based on the Hellmann–Feynman theorem. We use the norm-conserving pseudopotential proposed by Troullier and Martins [9] for the phosphorus atom. The s- and p-components of the pseudopotential are employed; the s-component is chosen to be a local term and the non-local p-component is treated by the Kleinman–Bylander method [10]. The exchange correlation energy is calculated in the local-density approximation [11,12]. The wavefunction, sampled at only C-point of the Brillouin zone, is expanded in a plane-wave basis set and their cutoff energy is 15 Ry, which is chosen so as to converge the total energy to within 1 mRy/electron. A cubic
Fig. 2. (a) The velocity autocorrelation function Z(t) normalized to Z(0). ^ (b) The spectrum of the Z(t), ZðxÞ. Three arrows indicate the frequencies observed for gas phase P4 (see text).
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Fig. 3. The dynamic structure factors, S(k, x) spectra normalized to S(k) for the low-density liquid (a) and high-density liquid (b). Curves at different k-values are vertically shifted.
with qðk; tÞ ¼
Z
qðr; tÞ expðik rÞdr ¼
X
expðik rj ðtÞÞ:
j
Here qðr; tÞ and rj(t) are the number density and the position of jth atom at time t, respectively. The dynamic structure factor is obtained by the Fourier transform of the F(k, t) as Z 1 Sðk; xÞ ¼ F ðk; tÞ expðixtÞdt: 2p Fig. 3(a) and (b) shows thus calculated dynamic structure factors for the low-density and high-density liquid phosphorus, respectively. The phonon peaks of the dynamic ˚ 1 structure factors are seen in the low-k at k = 0.43 A 1 ˚ for low-density liquid and at k = 0.51 A for high-density liquid. The central peak at zero frequency of the S(k, x) has ˚ 1 for the low-density liquid and at maximum at k = 1.3 A 1 ˚ k = 2.05 A in the high-density liquid. Though k = ˚ 1 and 0.51 A ˚ 1 are the shortest k-values related to 0.43 A 2p/(a side of cubic simulation box), the structure factors shown in Fig. 1 are in good agreement with experiment at these k-values and so S(k, x) at these k-values are reliable. 4. Discussion As shown in the previous studies [2,3], the low-density liquid is molecular liquid composed of P4 molecules. It is seen from Fig. 2(b) that the peaks in the spectrum at 42, 53, 70 meV of the molecular liquid correspond to the frequencies of the vibration modes of the P4 molecule, though it slightly shifts to the lower frequencies than that of the P4
molecule due to the intermolecular interaction in the liquid phase. The shoulder at x = 40 meV of the high-density liquid corresponds to the frequencies of the vibration of the mean distance between phosphorus atoms in the polymeric liquid, where the bond breaking and bond rearrangements occur frequently [3], and such unstable bond nature leads to broad peak, instead of a clear sharp peak. The widths of the central peak of the dynamic structure factor become narrower at wavenumbers close to the position of the main peak of S(k). This effect is called de Gennes narrowing and has its origins in the strong spatial correlations with these wavenumbers. The de Gennes nar˚ 1 for the low-density liquid rowing is seen at k = 1.3 A as shown in Fig. 3(a), which corresponds to the wavenumber of the first sharp diffraction peak (FSDP) of S(k) as seen in Fig. 1. It has been shown by our previous study [3] that the FSDP comes from the correlation between the P4 molecules in the low-density liquid. The narrowing of the central peak of S(k, x) for the high-density liquid ˚ 1, which coincides with is seen in Fig. 3(b) at k = 2.2 A k-value of the first peak position of S(k). The first peak comes from the second neighbor correlation in the complex polymeric high-density liquid [3]. In the present study, phonon peaks are confirmed at long wavelengths, which are seen in Fig. 3(a) at k = ˚ 1 for the low-density liquid and in Fig. 3(b) at 0.43 A ˚ 1 for the high-density liquid. These phonon k = 0.51 A peaks are not seen in higher-k region due to strong damping effects. To obtain the position of the phonon peak, we calculate the spectrum of the longitudinal-current correlation function Cl(k, x) defined by C l ðk; xÞ ¼ x2 Sðk; xÞ:
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the densities of two liquids, q = 1.7 g cm3 and q = 2.8 g cm3, and also from the different types of bonding, i.e. the molecular and polymeric liquids. 5. Conclusion We have performed ab initio molecular-dynamics simulations for the low-density and high-density liquid phosphorus to investigate their dynamical properties. We have shown that: (i) The vibration modes within the tetrahedral P4 molecules in the low-density molecular liquid can be seen in the calculated velocity autocorrelation function and its spectrum. (ii) A so-called de Gennes narrowing due to the strong spatial correlation between the P4 molecules and phonon peaks are seen in the calculated dynamic structure factors. (iii) The sound velocities estimated from the dispersion relation obtained from the dynamic structure factors are 2100 m s1 for the molecular liquid and 5300 m s1 for the polymeric liquid, which reflect the differences in the density and in the liquid state. Acknowledgments Fig. 4. The spectra of the longitudinal current correlation function Cl(k, x) for the low-density liquid and high-density liquid.
This research is supported by the Ministry of Education, Science, Sports and Culture, Japan, Grant-in-Aid for Scientific Research (No. 50324067). Calculations were performed using the facilities of the Computer Center of Kyushu University, the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo, and the supercomputers of Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA). References
Fig. 5. Dispersion relation for the low-density liquid (solid circles) and for the high-density liquid (open circles). The dashed lines are drawn for the estimation for the dispersion of the hydrodynamic sound.
In Fig. 4 we show the calculated Cl(k, x) for the lowdensity and high-density liquid phosphorus. The maximum peak of the Cl(k, x) in the low-k region corresponds to the phonon peak of the S(k, x). From the position of the phonon peak, we obtain the xk dispersion relations as shown in Fig. 5. The slope of the dispersion relation in the low-k and low-x limits gives the adiabatic sound velocity. The sound velocities of the liquids are roughly estimated to be about 2100 and 5300 m s1 for the low- and high-density liquid, respectively. The sound velocity of the polymeric liquid is two or three times faster than that of the molecular liquid. This result can be expected from a large difference in
[1] Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, K. Funakoshi, Nature 403 (2000) 170. [2] T. Morishita, Phys. Rev. Lett. 87 (2001) 105701. [3] Y. Senda, F. Shimojo, K. Hoshino, J. Phys.: Condens. Matter 14 (2002) 3715. [4] S. Hosokawa, J. Greif, F. Demmel, W.-C. Pilgrim, Nucl. Instr. Meth. B 199 (2003) 161. [5] S. Hosokawa, Y. Kawakita, W.-C. Pilgrim, H. Sinn, Phys. Rev. B 63 (2001) 134205. [6] Y. Kawakita, S. Hosokawa, T. Enosaki, K. Ohshima, S. Takeda, W.C. Pilgrim, S. Tsutsui, Y. Tanaka, A.Q.R. Baron, J. Phys. Soc. Jpn. 72 (2003) 1603. [7] T. Scopigno, G. Ruocco, F. Sette, Rev. Mod. Phys. 77 (2005) 881. [8] F. Shimojo, Y. Zempo, K. Hoshino, M. Watabe, Phys. Rev. B 52 (1995) 9320. [9] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [10] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) 425. [11] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [12] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [13] Y. Katayama, Y. Inamura, T. Mizutani, M. Yamakata, W. Utsumi, O. Shimomura, Science 36 (2004) 848. [14] Y.M. Bosworth, R.J.H. Clark, D.M. Rippon, J. Mol. Spectrosc. 46 (1973) 240. [15] A. Kornath, A. Kaufmann, M. Torheyden, J. Chem. Phys. 116 (2002) 3323.
Dynamical properties of liquid phosphorus studied by ab initio molecular-dynamics simulations Yasuhiro Senda
a,*
, Fuyuki Shimojo b, Kozo Hoshino
c
a
c
Department of Computational Science, Kanazawa University, Kanazawa 920-1192, Japan b Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan Available online 27 July 2007
Abstract We have performed ab initio molecular-dynamics simulations for the low-density and high-density liquid phosphorus to investigate their dynamical properties. We have shown that: (i) The vibration modes within the tetrahedral P4 molecules in the low-density molecular liquid can be seen in the calculated velocity autocorrelation function and its spectrum. (ii) A so-called de Gennes narrowing due to the strong spatial correlation between the P4 molecules and phonon peaks are seen in the calculated dynamic structure factors. (iii) The sound velocities estimated from the dispersion relation obtained from the dynamic structure factors are 2100 m s1 for the molecular liquid and 5300 m s1 for the polymeric liquid, which reflect the differences in the density and in the liquid state. Ó 2007 Published by Elsevier B.V. PACS: 61.20.Ja; 61.20.Lc Keywords: Acoustic properties and phonons; Liquid alloys and liquid metals; Diffraction and scattering measurements; Modeling and simulation; Ab initio; Molecular-dynamics; Structure
1. Introduction It is well known that phosphorus (P) shows a variety of structures for a wide range of pressures and temperatures; a molecular solid, the so-called ‘white’ phosphorus, is composed of tetrahedral P4 clusters, a variety of ‘red’ phosphorus that are usually amorphous, and an orthorhombic structure of ‘black’ phosphorus that consists of puckered double layers of sixfold rings. Katayama et al. [1] reported an X-ray diffraction observation of an abrupt pressure-induced structural change in liquid phosphorus at about 1 GPa as shown in Fig. 1. The transformation was expected to be a first-order liquid–
* Corresponding author. Present address: Graduate School of Science and Engineering, Yamaguchi University, Ube 755-8611, Japan. Tel.: +81 836 85 9812; fax: +81 836 85 9800. E-mail address: [email protected] (Y. Senda).
0022-3093/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.jnoncrysol.2007.05.102
liquid structural phase transition, accompanied by a discontinuous change of density of the liquid phosphorus. It was clearly shown by ab initio molecular-dynamics (MD) simulations [2,3] that the structural phase transition found by Katayama et al. [1] corresponds to the structural change from the molecular liquid composed of stable tetrahedral P4 molecules to the polymeric liquid with complex network structure. The calculated structure factors are in good agreement with those obtained by the X-ray diffraction experiments as shown in Fig. 1 and the characteristic features of the observed S(k) of liquid phosphorus for lower and higher pressures (densities) are well reproduced by ab initio MD simulations. It was also found that the structural change gives rise to the nonmetal–metal transition [3]. It is expected from these results for liquid phosphorus that the dynamical properties of the low-density and highdensity liquid phosphorus are also quite different. Though recent progress in inelastic X-ray scattering experiments
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supercell is used and periodic boundary conditions are imposed. The total number of atoms in the supercell is taken to be 100. As an initial configuration, P4 tetrahedra are arranged on the simple-cubic lattice. A constant-temperature MD simulation is carried out using the Nose´Hoover thermostat technique at the temperature of 1350 K. The observed densities [1,13] are used in our calculation. Since long time simulation is required for the determination of reliable dynamical quantities, we carry out our simulation for 24 ps with a time step of 2.4 fs after the thermal equilibrium state is achieved. 3. Results
Fig. 1. The S(k) for the liquid phosphorus at low-density (circles) and at high-density (squares). The full-line shows the experimental results at the corresponding pressures.
enables us to observe the dynamical properties of some liquids at high temperatures [4–7], little is known so far about the dynamical properties of the liquid phosphorus. In the present study, we carry out ab initio MD simulations for low-density and high-density liquid phosphorus to investigate the dynamical properties of them. We obtain the velocity autocorrelation functions and the dynamic structure factors and estimate the velocity of sound for these liquid phosphorus to discuss the dynamical properties of them comparatively.
The calculated velocity autocorrelation function Z(t), ^ and its spectrum ZðxÞ are shown in Fig. 2(a) and (b), respectively. As shown in Fig. 2(a), Z(t) of the low-density molecular liquid show an oscillating behaviour, which is related to the intramolecular vibration. It is shown in Fig. 2(b) that the low-density liquid has three peaks at x = 42, 53, ^ 70 meV in the ZðxÞ besides the diffusive mode in low-x region, while the high-density liquid has a shoulder at x = 40 meV. The frequencies of the P4 molecule in gas phase, x = 45.9, 57.9, 76.0 meV, observed by the Raman spectroscopy [14,15] are also indicated by arrows in Fig. 2(b). We have obtained by our simulation the intermediate scattering function F(k, t) defined by F ðk; tÞ ¼
1 hqðk; tÞqðk; 0Þi N
2. Method of calculation To study the structure and the electronic states of liquid phosphorus, we carry out an ab initio MD simulation, which is based on the density functional theory in the local-density approximation, the pseudopotential theory and the adiabatic approximation. In our calculation, we minimize the Kohn–Sham energy functional for a given ionic configuration at each time step by the conjugate-gradient method [8] and calculate the electron density and the forces acting on ions based on the Hellmann–Feynman theorem. We use the norm-conserving pseudopotential proposed by Troullier and Martins [9] for the phosphorus atom. The s- and p-components of the pseudopotential are employed; the s-component is chosen to be a local term and the non-local p-component is treated by the Kleinman–Bylander method [10]. The exchange correlation energy is calculated in the local-density approximation [11,12]. The wavefunction, sampled at only C-point of the Brillouin zone, is expanded in a plane-wave basis set and their cutoff energy is 15 Ry, which is chosen so as to converge the total energy to within 1 mRy/electron. A cubic
Fig. 2. (a) The velocity autocorrelation function Z(t) normalized to Z(0). ^ (b) The spectrum of the Z(t), ZðxÞ. Three arrows indicate the frequencies observed for gas phase P4 (see text).
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Fig. 3. The dynamic structure factors, S(k, x) spectra normalized to S(k) for the low-density liquid (a) and high-density liquid (b). Curves at different k-values are vertically shifted.
with qðk; tÞ ¼
Z
qðr; tÞ expðik rÞdr ¼
X
expðik rj ðtÞÞ:
j
Here qðr; tÞ and rj(t) are the number density and the position of jth atom at time t, respectively. The dynamic structure factor is obtained by the Fourier transform of the F(k, t) as Z 1 Sðk; xÞ ¼ F ðk; tÞ expðixtÞdt: 2p Fig. 3(a) and (b) shows thus calculated dynamic structure factors for the low-density and high-density liquid phosphorus, respectively. The phonon peaks of the dynamic ˚ 1 structure factors are seen in the low-k at k = 0.43 A 1 ˚ for low-density liquid and at k = 0.51 A for high-density liquid. The central peak at zero frequency of the S(k, x) has ˚ 1 for the low-density liquid and at maximum at k = 1.3 A 1 ˚ k = 2.05 A in the high-density liquid. Though k = ˚ 1 and 0.51 A ˚ 1 are the shortest k-values related to 0.43 A 2p/(a side of cubic simulation box), the structure factors shown in Fig. 1 are in good agreement with experiment at these k-values and so S(k, x) at these k-values are reliable. 4. Discussion As shown in the previous studies [2,3], the low-density liquid is molecular liquid composed of P4 molecules. It is seen from Fig. 2(b) that the peaks in the spectrum at 42, 53, 70 meV of the molecular liquid correspond to the frequencies of the vibration modes of the P4 molecule, though it slightly shifts to the lower frequencies than that of the P4
molecule due to the intermolecular interaction in the liquid phase. The shoulder at x = 40 meV of the high-density liquid corresponds to the frequencies of the vibration of the mean distance between phosphorus atoms in the polymeric liquid, where the bond breaking and bond rearrangements occur frequently [3], and such unstable bond nature leads to broad peak, instead of a clear sharp peak. The widths of the central peak of the dynamic structure factor become narrower at wavenumbers close to the position of the main peak of S(k). This effect is called de Gennes narrowing and has its origins in the strong spatial correlations with these wavenumbers. The de Gennes nar˚ 1 for the low-density liquid rowing is seen at k = 1.3 A as shown in Fig. 3(a), which corresponds to the wavenumber of the first sharp diffraction peak (FSDP) of S(k) as seen in Fig. 1. It has been shown by our previous study [3] that the FSDP comes from the correlation between the P4 molecules in the low-density liquid. The narrowing of the central peak of S(k, x) for the high-density liquid ˚ 1, which coincides with is seen in Fig. 3(b) at k = 2.2 A k-value of the first peak position of S(k). The first peak comes from the second neighbor correlation in the complex polymeric high-density liquid [3]. In the present study, phonon peaks are confirmed at long wavelengths, which are seen in Fig. 3(a) at k = ˚ 1 for the low-density liquid and in Fig. 3(b) at 0.43 A ˚ 1 for the high-density liquid. These phonon k = 0.51 A peaks are not seen in higher-k region due to strong damping effects. To obtain the position of the phonon peak, we calculate the spectrum of the longitudinal-current correlation function Cl(k, x) defined by C l ðk; xÞ ¼ x2 Sðk; xÞ:
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the densities of two liquids, q = 1.7 g cm3 and q = 2.8 g cm3, and also from the different types of bonding, i.e. the molecular and polymeric liquids. 5. Conclusion We have performed ab initio molecular-dynamics simulations for the low-density and high-density liquid phosphorus to investigate their dynamical properties. We have shown that: (i) The vibration modes within the tetrahedral P4 molecules in the low-density molecular liquid can be seen in the calculated velocity autocorrelation function and its spectrum. (ii) A so-called de Gennes narrowing due to the strong spatial correlation between the P4 molecules and phonon peaks are seen in the calculated dynamic structure factors. (iii) The sound velocities estimated from the dispersion relation obtained from the dynamic structure factors are 2100 m s1 for the molecular liquid and 5300 m s1 for the polymeric liquid, which reflect the differences in the density and in the liquid state. Acknowledgments Fig. 4. The spectra of the longitudinal current correlation function Cl(k, x) for the low-density liquid and high-density liquid.
This research is supported by the Ministry of Education, Science, Sports and Culture, Japan, Grant-in-Aid for Scientific Research (No. 50324067). Calculations were performed using the facilities of the Computer Center of Kyushu University, the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo, and the supercomputers of Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA). References
Fig. 5. Dispersion relation for the low-density liquid (solid circles) and for the high-density liquid (open circles). The dashed lines are drawn for the estimation for the dispersion of the hydrodynamic sound.
In Fig. 4 we show the calculated Cl(k, x) for the lowdensity and high-density liquid phosphorus. The maximum peak of the Cl(k, x) in the low-k region corresponds to the phonon peak of the S(k, x). From the position of the phonon peak, we obtain the xk dispersion relations as shown in Fig. 5. The slope of the dispersion relation in the low-k and low-x limits gives the adiabatic sound velocity. The sound velocities of the liquids are roughly estimated to be about 2100 and 5300 m s1 for the low- and high-density liquid, respectively. The sound velocity of the polymeric liquid is two or three times faster than that of the molecular liquid. This result can be expected from a large difference in
[1] Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, K. Funakoshi, Nature 403 (2000) 170. [2] T. Morishita, Phys. Rev. Lett. 87 (2001) 105701. [3] Y. Senda, F. Shimojo, K. Hoshino, J. Phys.: Condens. Matter 14 (2002) 3715. [4] S. Hosokawa, J. Greif, F. Demmel, W.-C. Pilgrim, Nucl. Instr. Meth. B 199 (2003) 161. [5] S. Hosokawa, Y. Kawakita, W.-C. Pilgrim, H. Sinn, Phys. Rev. B 63 (2001) 134205. [6] Y. Kawakita, S. Hosokawa, T. Enosaki, K. Ohshima, S. Takeda, W.C. Pilgrim, S. Tsutsui, Y. Tanaka, A.Q.R. Baron, J. Phys. Soc. Jpn. 72 (2003) 1603. [7] T. Scopigno, G. Ruocco, F. Sette, Rev. Mod. Phys. 77 (2005) 881. [8] F. Shimojo, Y. Zempo, K. Hoshino, M. Watabe, Phys. Rev. B 52 (1995) 9320. [9] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [10] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) 425. [11] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [12] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [13] Y. Katayama, Y. Inamura, T. Mizutani, M. Yamakata, W. Utsumi, O. Shimomura, Science 36 (2004) 848. [14] Y.M. Bosworth, R.J.H. Clark, D.M. Rippon, J. Mol. Spectrosc. 46 (1973) 240. [15] A. Kornath, A. Kaufmann, M. Torheyden, J. Chem. Phys. 116 (2002) 3323.