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The temperature dependent collective dynamics of liquid Rubidium
Journal of Non-Crystalline Solids 353 (2007) 3164–3168 www.elsevier.com/locate/jnoncrysol
The temperature dependent collective dynamics of liquid Rubidium F. Demmel b
a,*
, A. Diepold b, H. Aschauer b, C. Morkel
b
a ILL, B.P. 156, F-38042 Grenoble cedex 9, France TU Mu¨nchen, Physikdepartment E21, D-85747 Garching, Germany
Received 11 August 2004 Available online 17 July 2007
Abstract Liquids alkali metals show near the melting point at small momentum transfer values an upward bending of the dispersion relation. This so-called positive dispersion can be described within generalised hydrodynamics as a visco-elastic reaction of the liquid. There is a speculation that long-living clusters could be the physical reason behind this phenomenon. To shed light on this question a coherent inelastic neutron scattering experiment on liquid Rubidium was performed at four temperatures starting at the melting point. Distinct deviations from molecular dynamics simulation results occur at the lowest accessible momentum transfer values. With rising temperature the excitation frequencies soften at a momentum transfer value which is correlated with the dimensions of the supposed clusters. 2007 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 61.25.Mv; 63.50.+x Keywords: Liquid alloys and liquid metals; Diffraction and scattering measurements; Modeling and simulation
1. Introduction The study of the collective dynamics of liquid alkali metals has a long tradition, because they show distinct inelastic excitations. In particular, liquid Rubidium is a well studied model system due to its favourable scattering properties for neutrons. The pioneering experiment in inelastic collective dynamics was done about 30 years ago by Copley and Rowe on liquid Rubidium [1]. Later on the collective dynamics has been measured for a lot of thermodynamic states, even up to the critical point [2]. One common result for the derived dispersion relations of liquid Rubidium and also other alkali metals near the melting point was the socalled ‘positive dispersion’, an upward bending of the dispersion above the values expected from adiabatic sound velocity. This upward bending can be understood as a visco-elastic reaction of the liquid. As the frequency rises
*
Corresponding author. Present address: ISIS Facility, Rutherford Appleton Laboratory, Didcot, OX11 0QX, UK. Fax: +44 1235 445103. E-mail address: [email protected] (F. Demmel). 0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.05.050
above a certain relaxation time value the liquid responses in a solid-like manner. There is a speculation that this solid-like feature in liquid dynamics is reasoned from temporarily staying clusters in the liquid near the melting point [3]. A half century ago Frank has calculated that an icosahedron of Lennard– Jones atoms is energetically more stable than the crystalline counterpart [4]. There is now an active research whether clusters can be found in the supercooled and liquid state. Evidences are coming from X-ray diffraction at the liquid surface where a five fold symmetry was found [5]. Furthermore levitation experiments allow to access the supercooled state where supposed clusters should appear more pronounced. An analysis including icosahedral structures support this view [6]. Our intention is focused on possible influences of clusters onto the collective dynamics of simple liquids. Interestingly the diameter of an icosahedron corresponds to the measured maximum of the positive dispersion in reciprocal space [3]. If these clusters exist, then the question arises whether they live infinite up to the boiling point or they melt at a certain temperature. The melting of clusters would define a novel
F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
temperature T * which would signal the start of solidification during cooling on a microscopic length scale. With a simple approximation we have estimated the possible temperature when this crossover can happen in liquid Rubidium. The high frequency sound velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 ðQ ! 0Þ ¼
3kT M
þ 103 x2E r2 is a good measure for the
maximal experimental results. With melting of the clusters the positive dispersion should vanish. For this we assume that at this temperature the solid-like sound velocity c1 will be equal to the adiabatic value c 2 c1 ðQ ! 0Þ 3v2 þ 0:3x2 r2 ¼1¼ 0 2 c E : c v0 Sð0Þ The thermal velocity v0 = kBT/M is related with the temperature T and the other parameters are treated temperature independent. If we solve the formula to get T and put into the formula for the Einstein frequency xE = 6.1 · 1012 s1, for the specific heat ratio c = 1.1 ˚ , we get for and for the hard sphere parameter r = 4.4 A the temperature T* = 457 K. More generally formulated, we look for the temperature when the kinetic energy term equalizes the potential energy part. The purpose of our investigation was twofold. At first, there exists a dedicated MD-simulation on liquid Rubidium for the same Q-range and temperatures [7]. This opens up the possibility to compare as direct as possible experimental results with a simulation. Secondly we will look for temperature dependent changes in the inelastic peak positions at certain Q-vectors, which correspond to the supposed dimensions of the icosahedrons. Former experimental investigations on liquid Rubidium or liquid Sodium have focused on different momentum transfer regions or with larger temperature steps [8,9]. 2. Experiment The coherent dynamic structure factor S(Q, x) of liquid Rubidium has been measured with neutrons. In a water and oxygen free glove box Rubidium was filled into a cylindrical Aluminium can with 16 mm diameter (9% scatterer). The cell was then electron beam welded. Inelastic neutron scattering measurement have been performed at the Forschungsreaktor Mu¨nchen FRM with a multi analyser three axis spectrometer [10]. With this novel type of instrument 30 scattering angles could be measured simultaneously in a nearly constant Q mode. The energy resolution was ˚ 1. Momentum trans1.3 meV and the Q resolution ±0.03 A ˚ 1 up to Q = 2.5 A ˚ 1 fer values are ranging from Q = 0.43 A and therefore well beyond the structure factor maximum. Four temperatures have been measured: 315 K, 412 K, 503 K, 600 K (melting temperature of Rubidium Tmelt = 312.65 K). The energy transfers range from 2 meV up to 9.5 meV. Several scans have been performed for each temperature. The sample was kept in a vacuum box to avoid air scattering. Heating was performed by two resistance heaters on top and bottom of the cell. The tempera-
3165
ture uncertainty was smaller 1.5 K during all experiments. No second order filter was installed at this time and it is known that the second order contribution is about 15% at the chosen end energy of 31 meV. The second order contribution spoils the spectra at momentum transfer values of half the value of the structure factor maximum with a superposed further quasielastic line. The evaluation of the data included an empty cell subtraction with absorption correction according to Paalman/Pings [11]. Absolute calibration has been performed with a vanadium standard. All spectra have been symmetrised. Multiple scattering has been treated with a new developed program. This program is an extension of the formalism for elastic incoherent scattering by Blech and Averbach [12]. A simulation program has been developed that calculates two times scattered neutrons in an efficient way for a prescribed single scattering law. This is performed by numerical evaluation of the analytically exact secondary scattering integral, regarding the proper kinematic restrictions. As in a suitable neutron scattering experiment the scattering power of a sample is about 10% of the incident intensity, secondary scattered intensity is about 1%, leaving roughly a multiple scattering intensity of the order of 0.1%, which is neglected by the program and is well beyond the statistical error of ordinary neutron scattering data. As an input scattering law the coherent Lovesey model was used which describes the spectra reasonable well over a large range of Q-vectors [13]. The data analysis was made in the same way as it was done with the MD simulation data [7]. Therefore we extracted from the measured spectra resolution deconvoluted current spectra. The deconvolution assures that we do not get a shift in the peak positions due to resolution broadening in particular at the smallest Q-vectors. As a model function to parametrise the spectra we used the coherent Lovesey model [13] SðQ; xÞ ¼
SðQÞ x20 ðx2l x20 Þs : p ðxsðx2 x2l ÞÞ2 þ ðx2 x20 Þ2
To get resolution deconvoluted results the S(Q, x) spectra have been fitted to this model function. All four parameters S(Q), s, x02 and xl2 have been varied to get the best description of the spectra. Nevertheless, the fit values for S(Q) and x02 are generally near the values calculated from the moments except at the structure factor maximum. From the deconvoluted spectra current spectra j(Q, x) = x2S(Q, x) have been calculated. The current spectra show for all Q-vectors an inelastic peak which can be used to define a dispersion relation xpeak(Q). The comparison with the MD data is made then via wave vector dependent phase velocities cðQÞ ¼ xpeak ðQÞ=Q. 3. Results Fig. 1 shows the spectra of the four smallest Q-vectors for the two extreme temperatures 315 K and 600 K. Clearly
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F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
Fig. 1. Measured spectra for the four smallest Q-vectors are shown. On the left side the spectra near the melting point and on the right side at our highest temperature 600 K. The line is a guide for the eye.
the inelastic excitations are visible at the lower temperature and also at the higher temperature. At higher temperature the spectra broaden but the inelastic excitations are still visible. In Fig. 2 the velocities deduced from peak positions of the current spectra are shown for the four temperatures in comparison with the results from the MD-simulation. Included as a line is furthermore the calculated high frequency sound velocity c1 ðQ ! 0Þ. This quantity is related to the derivatives of the potential and has been calculated for the different temperatures [7]. The absolute values at Q-vectors around the structure factor maximum are in good agreement with results from Chieux et al. [9]. At small Q-vectors our results at the melting temperature are slightly higher compared to the dispersion from Copley and Rowe [2], but fits well with data reported in [3]. This fact could be related to different methods in evaluating the inelastic peak positions. It has been shown that using peaks in the current spectra can give higher values than fit models directly applied to the spectra [9]. In Fig. 3 differences of phase velocities from the neutron measurements are shown. The error bars are deduced from typical fit errors which amount to 2–5% of the peak frequency for our statistical quality. As a conservative estimate we have put the larger value as error bar.
4. Discussion Fig. 1 shows that even at a temperature as high as 2 · Tm inelastic excitations are visible in the spectra. There is a good agreement between the experiment and the simulation over nearly the whole explored Q-range concerning the peak positions of the current spectra for all four investigated temperatures (see Fig. 2). At the lowest momentum ˚ 1 the experimental valtransfer vectors around Q = 0.5 A ues deviate to higher velocities. At small Q-vectors the excitations are governed by the potential part of the energy. This deviation can be a hint that the used potential is not able to reproduce all dynamic features of liquid Rubidium. With increasing temperature the experimental values soften and approach the MD-simulation values at these small Q-vectors. Regarding the changes within the experimental values there is a marked change in the phase velocities at ˚ 1 around 450 K (see Fig. 3). Although the total Q = 0.45 A decrease is only about 15% it is well beyond the error bars. This frequency decrease in a certain momentum transfer region could be interpreted as a break down of solid-like features in the liquid state. Even in the spectra of Fig. 1 one can already identify a shift in the positions of the inelastic excitations at the two smallest Q-vectors. This
F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
3167
Fig. 2. The deduced velocities from the inelastic peaks of the current spectra are show for the four temperatures (squares). Included are the results from the MD-simulation (triangles). The different curves are connected by a line as a guide for the eye. The solid line at small Q-vectors shows the calculated infinite frequency velocity c1 ðQ ! 0Þ.
or equivalent of frequency. This behavior is in accordance with results from [9] and reflects the transition to the free particle regime for large Q-vectors. 5. Conclusion
Fig. 3. The velocity differences between the experimental results are plotted for the three temperature differences and connected by a line as a guide for the eye.
change with temperature is independent of the fit model as an evaluation with a different model function (damped harmonic oscillator) gives the same result. For large Q-vectors one observes with rising temperature an increase of velocity
An inelastic neutron scattering experiment has been performed on liquid Rubidium. The data analysis was done in the way as for a previously performed MD-simulation to compare as precise as possible the results. There is a good agreement between the experimental results and the MDsimulation, except at the lowest Q-vectors. The inelastic peak positions in the current spectra of the experiment show a frequency decrease between 400 K and 500 K at momentum transfer vectors which correspond to dimensions of the supposed icosahedral clusters. This result could be interpreted as melting of solid-like structures in liquid Rubidium. References [1] J.R.D. Copley, J.M. Rowe, Phys. Rev. Lett. 32 (1974) 49. [2] W.C. Pilgrim, M. Ross, L.H. Yang, F. Hensel, Phys. Rev. Lett. 78 (1997) 3685.
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F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
[3] C. Morkel, T. Bodensteiner, H. Gemperlein, Phys. Rev. E 47 (1993) 2575. [4] F.C. Frank, Proc. R. Soc. London A 215 (1952) 43. [5] H. Reichert, O. Klein, H. Dosch, M. Denk, V. Honkimaki, T. Lippmann, G. Reiter, Nature 408 (2000) 839. [6] T. Schenk, D. Holland-Moritz, V. Simonet, R. Bellissent, D.M. Herlach, Phys. Rev. Lett. 89 (2002) 075507-1. [7] D. Pasqualini, R. Vallauri, F. Demmel, Chr. Morkel, U. Balucani, J. Non-Cryst. Solids 250–252 (1999) 76.
[8] W.C. Pilgrim, S. Hosokawa, H. Saggau, H. Sinn, E. Burkel, J. NonCryst. Solids 250–252 (1999) 96. [9] P. Chieux, J. Dupuy-Philon, J. Jal, J.B. Suck, J. Non-Cryst. Solids 205–207 (1996) 370. [10] F. Demmel, A. Fleischmann, W. Gla¨ser, NIM A 416 (1998) 115. [11] H.H. Paalman, C.J. Pings, J. Appl. Phys. 33 (1962) 2635. [12] I. Blech, B. Averbach, Phys. Rev. 137 (1965) 1113. [13] S.W. Lovesey, J. Phys. C 4 (1971) 3057.
The temperature dependent collective dynamics of liquid Rubidium F. Demmel b
a,*
, A. Diepold b, H. Aschauer b, C. Morkel
b
a ILL, B.P. 156, F-38042 Grenoble cedex 9, France TU Mu¨nchen, Physikdepartment E21, D-85747 Garching, Germany
Received 11 August 2004 Available online 17 July 2007
Abstract Liquids alkali metals show near the melting point at small momentum transfer values an upward bending of the dispersion relation. This so-called positive dispersion can be described within generalised hydrodynamics as a visco-elastic reaction of the liquid. There is a speculation that long-living clusters could be the physical reason behind this phenomenon. To shed light on this question a coherent inelastic neutron scattering experiment on liquid Rubidium was performed at four temperatures starting at the melting point. Distinct deviations from molecular dynamics simulation results occur at the lowest accessible momentum transfer values. With rising temperature the excitation frequencies soften at a momentum transfer value which is correlated with the dimensions of the supposed clusters. 2007 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 61.25.Mv; 63.50.+x Keywords: Liquid alloys and liquid metals; Diffraction and scattering measurements; Modeling and simulation
1. Introduction The study of the collective dynamics of liquid alkali metals has a long tradition, because they show distinct inelastic excitations. In particular, liquid Rubidium is a well studied model system due to its favourable scattering properties for neutrons. The pioneering experiment in inelastic collective dynamics was done about 30 years ago by Copley and Rowe on liquid Rubidium [1]. Later on the collective dynamics has been measured for a lot of thermodynamic states, even up to the critical point [2]. One common result for the derived dispersion relations of liquid Rubidium and also other alkali metals near the melting point was the socalled ‘positive dispersion’, an upward bending of the dispersion above the values expected from adiabatic sound velocity. This upward bending can be understood as a visco-elastic reaction of the liquid. As the frequency rises
*
Corresponding author. Present address: ISIS Facility, Rutherford Appleton Laboratory, Didcot, OX11 0QX, UK. Fax: +44 1235 445103. E-mail address: [email protected] (F. Demmel). 0022-3093/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.05.050
above a certain relaxation time value the liquid responses in a solid-like manner. There is a speculation that this solid-like feature in liquid dynamics is reasoned from temporarily staying clusters in the liquid near the melting point [3]. A half century ago Frank has calculated that an icosahedron of Lennard– Jones atoms is energetically more stable than the crystalline counterpart [4]. There is now an active research whether clusters can be found in the supercooled and liquid state. Evidences are coming from X-ray diffraction at the liquid surface where a five fold symmetry was found [5]. Furthermore levitation experiments allow to access the supercooled state where supposed clusters should appear more pronounced. An analysis including icosahedral structures support this view [6]. Our intention is focused on possible influences of clusters onto the collective dynamics of simple liquids. Interestingly the diameter of an icosahedron corresponds to the measured maximum of the positive dispersion in reciprocal space [3]. If these clusters exist, then the question arises whether they live infinite up to the boiling point or they melt at a certain temperature. The melting of clusters would define a novel
F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
temperature T * which would signal the start of solidification during cooling on a microscopic length scale. With a simple approximation we have estimated the possible temperature when this crossover can happen in liquid Rubidium. The high frequency sound velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 ðQ ! 0Þ ¼
3kT M
þ 103 x2E r2 is a good measure for the
maximal experimental results. With melting of the clusters the positive dispersion should vanish. For this we assume that at this temperature the solid-like sound velocity c1 will be equal to the adiabatic value c 2 c1 ðQ ! 0Þ 3v2 þ 0:3x2 r2 ¼1¼ 0 2 c E : c v0 Sð0Þ The thermal velocity v0 = kBT/M is related with the temperature T and the other parameters are treated temperature independent. If we solve the formula to get T and put into the formula for the Einstein frequency xE = 6.1 · 1012 s1, for the specific heat ratio c = 1.1 ˚ , we get for and for the hard sphere parameter r = 4.4 A the temperature T* = 457 K. More generally formulated, we look for the temperature when the kinetic energy term equalizes the potential energy part. The purpose of our investigation was twofold. At first, there exists a dedicated MD-simulation on liquid Rubidium for the same Q-range and temperatures [7]. This opens up the possibility to compare as direct as possible experimental results with a simulation. Secondly we will look for temperature dependent changes in the inelastic peak positions at certain Q-vectors, which correspond to the supposed dimensions of the icosahedrons. Former experimental investigations on liquid Rubidium or liquid Sodium have focused on different momentum transfer regions or with larger temperature steps [8,9]. 2. Experiment The coherent dynamic structure factor S(Q, x) of liquid Rubidium has been measured with neutrons. In a water and oxygen free glove box Rubidium was filled into a cylindrical Aluminium can with 16 mm diameter (9% scatterer). The cell was then electron beam welded. Inelastic neutron scattering measurement have been performed at the Forschungsreaktor Mu¨nchen FRM with a multi analyser three axis spectrometer [10]. With this novel type of instrument 30 scattering angles could be measured simultaneously in a nearly constant Q mode. The energy resolution was ˚ 1. Momentum trans1.3 meV and the Q resolution ±0.03 A ˚ 1 up to Q = 2.5 A ˚ 1 fer values are ranging from Q = 0.43 A and therefore well beyond the structure factor maximum. Four temperatures have been measured: 315 K, 412 K, 503 K, 600 K (melting temperature of Rubidium Tmelt = 312.65 K). The energy transfers range from 2 meV up to 9.5 meV. Several scans have been performed for each temperature. The sample was kept in a vacuum box to avoid air scattering. Heating was performed by two resistance heaters on top and bottom of the cell. The tempera-
3165
ture uncertainty was smaller 1.5 K during all experiments. No second order filter was installed at this time and it is known that the second order contribution is about 15% at the chosen end energy of 31 meV. The second order contribution spoils the spectra at momentum transfer values of half the value of the structure factor maximum with a superposed further quasielastic line. The evaluation of the data included an empty cell subtraction with absorption correction according to Paalman/Pings [11]. Absolute calibration has been performed with a vanadium standard. All spectra have been symmetrised. Multiple scattering has been treated with a new developed program. This program is an extension of the formalism for elastic incoherent scattering by Blech and Averbach [12]. A simulation program has been developed that calculates two times scattered neutrons in an efficient way for a prescribed single scattering law. This is performed by numerical evaluation of the analytically exact secondary scattering integral, regarding the proper kinematic restrictions. As in a suitable neutron scattering experiment the scattering power of a sample is about 10% of the incident intensity, secondary scattered intensity is about 1%, leaving roughly a multiple scattering intensity of the order of 0.1%, which is neglected by the program and is well beyond the statistical error of ordinary neutron scattering data. As an input scattering law the coherent Lovesey model was used which describes the spectra reasonable well over a large range of Q-vectors [13]. The data analysis was made in the same way as it was done with the MD simulation data [7]. Therefore we extracted from the measured spectra resolution deconvoluted current spectra. The deconvolution assures that we do not get a shift in the peak positions due to resolution broadening in particular at the smallest Q-vectors. As a model function to parametrise the spectra we used the coherent Lovesey model [13] SðQ; xÞ ¼
SðQÞ x20 ðx2l x20 Þs : p ðxsðx2 x2l ÞÞ2 þ ðx2 x20 Þ2
To get resolution deconvoluted results the S(Q, x) spectra have been fitted to this model function. All four parameters S(Q), s, x02 and xl2 have been varied to get the best description of the spectra. Nevertheless, the fit values for S(Q) and x02 are generally near the values calculated from the moments except at the structure factor maximum. From the deconvoluted spectra current spectra j(Q, x) = x2S(Q, x) have been calculated. The current spectra show for all Q-vectors an inelastic peak which can be used to define a dispersion relation xpeak(Q). The comparison with the MD data is made then via wave vector dependent phase velocities cðQÞ ¼ xpeak ðQÞ=Q. 3. Results Fig. 1 shows the spectra of the four smallest Q-vectors for the two extreme temperatures 315 K and 600 K. Clearly
3166
F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
Fig. 1. Measured spectra for the four smallest Q-vectors are shown. On the left side the spectra near the melting point and on the right side at our highest temperature 600 K. The line is a guide for the eye.
the inelastic excitations are visible at the lower temperature and also at the higher temperature. At higher temperature the spectra broaden but the inelastic excitations are still visible. In Fig. 2 the velocities deduced from peak positions of the current spectra are shown for the four temperatures in comparison with the results from the MD-simulation. Included as a line is furthermore the calculated high frequency sound velocity c1 ðQ ! 0Þ. This quantity is related to the derivatives of the potential and has been calculated for the different temperatures [7]. The absolute values at Q-vectors around the structure factor maximum are in good agreement with results from Chieux et al. [9]. At small Q-vectors our results at the melting temperature are slightly higher compared to the dispersion from Copley and Rowe [2], but fits well with data reported in [3]. This fact could be related to different methods in evaluating the inelastic peak positions. It has been shown that using peaks in the current spectra can give higher values than fit models directly applied to the spectra [9]. In Fig. 3 differences of phase velocities from the neutron measurements are shown. The error bars are deduced from typical fit errors which amount to 2–5% of the peak frequency for our statistical quality. As a conservative estimate we have put the larger value as error bar.
4. Discussion Fig. 1 shows that even at a temperature as high as 2 · Tm inelastic excitations are visible in the spectra. There is a good agreement between the experiment and the simulation over nearly the whole explored Q-range concerning the peak positions of the current spectra for all four investigated temperatures (see Fig. 2). At the lowest momentum ˚ 1 the experimental valtransfer vectors around Q = 0.5 A ues deviate to higher velocities. At small Q-vectors the excitations are governed by the potential part of the energy. This deviation can be a hint that the used potential is not able to reproduce all dynamic features of liquid Rubidium. With increasing temperature the experimental values soften and approach the MD-simulation values at these small Q-vectors. Regarding the changes within the experimental values there is a marked change in the phase velocities at ˚ 1 around 450 K (see Fig. 3). Although the total Q = 0.45 A decrease is only about 15% it is well beyond the error bars. This frequency decrease in a certain momentum transfer region could be interpreted as a break down of solid-like features in the liquid state. Even in the spectra of Fig. 1 one can already identify a shift in the positions of the inelastic excitations at the two smallest Q-vectors. This
F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
3167
Fig. 2. The deduced velocities from the inelastic peaks of the current spectra are show for the four temperatures (squares). Included are the results from the MD-simulation (triangles). The different curves are connected by a line as a guide for the eye. The solid line at small Q-vectors shows the calculated infinite frequency velocity c1 ðQ ! 0Þ.
or equivalent of frequency. This behavior is in accordance with results from [9] and reflects the transition to the free particle regime for large Q-vectors. 5. Conclusion
Fig. 3. The velocity differences between the experimental results are plotted for the three temperature differences and connected by a line as a guide for the eye.
change with temperature is independent of the fit model as an evaluation with a different model function (damped harmonic oscillator) gives the same result. For large Q-vectors one observes with rising temperature an increase of velocity
An inelastic neutron scattering experiment has been performed on liquid Rubidium. The data analysis was done in the way as for a previously performed MD-simulation to compare as precise as possible the results. There is a good agreement between the experimental results and the MDsimulation, except at the lowest Q-vectors. The inelastic peak positions in the current spectra of the experiment show a frequency decrease between 400 K and 500 K at momentum transfer vectors which correspond to dimensions of the supposed icosahedral clusters. This result could be interpreted as melting of solid-like structures in liquid Rubidium. References [1] J.R.D. Copley, J.M. Rowe, Phys. Rev. Lett. 32 (1974) 49. [2] W.C. Pilgrim, M. Ross, L.H. Yang, F. Hensel, Phys. Rev. Lett. 78 (1997) 3685.
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F. Demmel et al. / Journal of Non-Crystalline Solids 353 (2007) 3164–3168
[3] C. Morkel, T. Bodensteiner, H. Gemperlein, Phys. Rev. E 47 (1993) 2575. [4] F.C. Frank, Proc. R. Soc. London A 215 (1952) 43. [5] H. Reichert, O. Klein, H. Dosch, M. Denk, V. Honkimaki, T. Lippmann, G. Reiter, Nature 408 (2000) 839. [6] T. Schenk, D. Holland-Moritz, V. Simonet, R. Bellissent, D.M. Herlach, Phys. Rev. Lett. 89 (2002) 075507-1. [7] D. Pasqualini, R. Vallauri, F. Demmel, Chr. Morkel, U. Balucani, J. Non-Cryst. Solids 250–252 (1999) 76.
[8] W.C. Pilgrim, S. Hosokawa, H. Saggau, H. Sinn, E. Burkel, J. NonCryst. Solids 250–252 (1999) 96. [9] P. Chieux, J. Dupuy-Philon, J. Jal, J.B. Suck, J. Non-Cryst. Solids 205–207 (1996) 370. [10] F. Demmel, A. Fleischmann, W. Gla¨ser, NIM A 416 (1998) 115. [11] H.H. Paalman, C.J. Pings, J. Appl. Phys. 33 (1962) 2635. [12] I. Blech, B. Averbach, Phys. Rev. 137 (1965) 1113. [13] S.W. Lovesey, J. Phys. C 4 (1971) 3057.