Modified Landau–Placzek ratio of the liquid metal rubidium beyond hydrodynamics

The intensity ratio of the Rayleigh line and the Brillouin lines can be derived within hydrodynamics and is known as the Landau–Placzek (LP) ratio. This ratio is directly related to the ratio of specific heats of the fluid. Within the microscopic wave vector range, which can be probed by inelastic neutron scattering, the intensity ratio for simple liquid metals deviates distinctly from the hydrodynamic prediction of the LP-ratio. We derive the intensity ratio from experimental data of liquid rubidium, which shows an enhanced LP-ratio by a factor 8 compared to the hydrodynamic prediction. This strong deviation indicates a further relaxation process in the microscopic wave vector range beyond hydrodynamics. That relaxation process is the viscoelastic reaction of the simple liquid to density fluctuations. Taking this process into account a modified LP-ratio is able to describe the data quite well.


Introduction
A fluid in thermal equilibrium will permanently endure fluctuations, which will decay to the equilibrium state. On long length scales compared to the molecular distances and on long time scales compared to typical collision times these processes can be described through hydrodynamics. Applying linearized Navier-Stokes equations the decay of density fluctuations can be described as a flow process in this continuum description. The result for the lineshape of the density correlation function is a triplet for the collective excitation spectrum [1][2][3][4]. Density fluctuations are related to pressure and entropy fluctuations * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. within statistical mechanics, see for example [5,6]. The density fluctuations will decay through propagating sound modes, the Brillouin lines, and show a central line, the Rayleigh line, originating from entropy fluctuations. The sound modes disperse linearly with wave vector and the widths of all lines increase with a quadratic wave vector dependence. The ratio of the Rayleigh line intensity I R and the intensity of the Brillouin lines I B has been derived within hydrodynamics and is known as the Landau-Placzek (LP) ratio [7]: Here, γ denotes the ratio of the specific heats cP cV . Thermodynamics connects the specific heat ratio with the ratio of the isothermal and adiabatic compressibility, which are directly related to the adiabatic c s and isothermal velocity of sound c 0 . One condition for the application of the LP-ratio is that no other relaxation process occurs in the fluid on a similar time scale as the relaxation process for pressure and temperature [7]. With light scattering experiments the success of the hydrodynamic approach for a monatomic fluid, like liquid argon, could be demonstrated and the LP-ratio is fulfilled [8].
During the investigation of molecular fluids it was observed that the LP-ratio can not be accomplished anymore and it was argued that the relaxation by internal modes of the molecule will alter the LP-ratio [9]. This is not the only relaxation process which can change the LP-ratio. Also relaxation processes for the bulk viscosity, shear viscosity and structural relaxation have been considered to modify the LP-ratio [2]. More recently changes in the LP-ratio with temperature on glass forming molecular liquids have been recorded as a measure to identify a transition between different dynamic regimes [10]. Some of these relaxation processes can also occur in monatomic liquids and a deviation from the hydrodynamic LPratio is not restricted to complex liquids. Landau mentions the relaxation of elastic strain, which occurs with Maxwell's relaxation time τ M , as one of the fastest relaxation processes in a liquid as a possible reason for a deviation of the LP-ratio [7]. Maxwell's relaxation time is on the same time scale as the inverse excitation frequencies in a monatomic liquid in the microscopic wave vector range. In order to include further relation processes into the analysis various approaches have been proposed, see for example [9,[11][12][13].
For liquid alkali metals many inelastic measurements for the collective dynamics exist. The wealth of data is related to the fact that distinct inelastic excitations appear in the spectra over a wide range in wave vectors. Alkali metals have a specific heat ratio of about 1.1 and hence the central line should have a small intensity compared to the inelastic lines, according to the LP-ratio. However, this is in stark contrast to the spectra measured by inelastic neutron scattering and inelastic x-ray scattering [14][15][16][17][18]. The large central line indicates a further relaxation process in the explored wave vector range. Also molecular dynamics simulations indicate a similar trend [19,20] Here, we present experimental data from liquid rubidium near the melting point and derive the measured intensity ratio between the central line and the inelastic modes. A modified LP-ratio will then be applied to describe the experimental data.
Liquid rubidium has often been used to study the collective dynamics of a simple liquid metal due to its favorable neutron scattering properties. The negligible incoherent cross section, low absorption and a reasonable large coherent cross section makes rubidium to a preferred element for studying collective dynamics [21,22]. After the pioneering experiment by Copley and Rowe near the melting point liquid rubidium was used regularly to study collective dynamics with increasing wave vector and temperature [18,[23][24][25][26]. Also the fine changes in structural relaxation dynamics around the main structure factor peak were thoroughly investigated [27][28][29][30].

Experimental details
Most of the data presented here have been measured at the FRM reactor in Garching, Germany, on a multidetector three axis spectrometer [31,32]. The data have been fully corrected including a multiple scattering correction and all details of the experiment and data analysis can be found in the literature [18,26]. Due to the absence of a second order neutron filter during this experiment some Q-vectors around Q ≈ 0.75 Å −1 can not be used, because the central line is contaminated with scattering of second order neutrons from the strong structure factor peak. Toward smaller Q-vectors two scans at Q = 0.3 Å −1 and Q = 0.4 Å −1 have been added to the data analysis, because the previous experiment was limited to a minimum Q-vector of Q = 0.43 Å −1 . This additional measurement was performed with the IN3 spectrometer at the ILL, France. The same sample, an electron beam welded 16 mm diameter cylindrical aluminium can filled with rubidium, was used. The energy resolution was improved to ∆E = 0.78 meV compared to the ∆E = 1.35 meV from the previous experiment [18,31] and a tight collimation was applied to reach small scattering angles down to 1 0 . The sample was placed inside a vacuum box to reduce air scattering when reaching the small scattering angles. An empty can subtraction including properly calculated absorption factors and a multiple scattering correction was applied to obtain the final scattering data.
Liquid alkali metals show well-defined inelastic excitations near the melting point over a large range of wave vectors [14,18]. To describe the full lineshape over a wide wave vector range and over a wide range in temperatures applying a memory function approach with a single or even two exponential decaying memory functions has proved to be successful [18,33]. Within generalized hydrodynamics a different approach uses a sum of complex Lorentzian functions to describe the collective dynamics [34,35].
The well-defined spectra near the melting point allow to describe the spectra with a simple model in order to separate the individual contributions to the full lineshape. To achieve this aim we fitted a sum of two Lorentzians convoluted with the respective Gaussian resolution function to the measured spectra, which covered the Rayleigh line and one Brillouin mode. This procedure is robust and the most simple approach to determine the necessary information, namely the intensity ratios, in our case. Figure 1 shows three spectra of liquid rubidium taken near the melting point. The spectra demonstrate inelastic excitations which are well separated from the central line at small wave vectors. At the largest wave vector Q = 1.0 Å −1 both lines broaden and approach each other. From that wave vector onwards an unambiguous separation of both spectral lines becomes more and more difficult. The spectrum at Q = 0.3 Å −1 has been measured separately and the statistics is worse compared to all the other spectra due to the more tight collimation and better energy resolution. Included in the plots are the total fits with two Lorentzian lines convoluted with the respective resolution function and the individual spectral components as dashed lines. Please note that only the side with positive energy transfers has been modeled by an inelastic Lorentzian. Consequently the fit (line) appears visually not symmetric around the quasielastic line, which, however, has no consequences for the obtained intensity ratios. Within this wave vector range the fit with two Lorentzians provides a reasonable good description of the line shape over the whole used wave vector range and we can expect to get reliable intensity values. With the specific heat ratio γ = 1.1 [36] for liquid rubidium the central line would have about 10% intensity compared to the total inelastic mode intensity. Obviously this is not the case here. The strong central line indicates that a further relaxation process is at work, which then results in a modification to the LP-ratio.

Results and discussion
Mountain considered the case of a molecular fluid when a further internal relaxation process appears within the hydrodynamic description [9]. He assumed that the bulk viscosity will have an exponentially decaying component in time, which corresponds to frequency dependent transport coefficients entering the hydrodynamic description [12]. The calculated spectral lineshape consists now of two central lines and two Brillouin modes. As a consequence this further relaxation process will shift the phonon modes from the adiabatic value c s to a high-frequency sound velocity c ∞ . The LP-ratio is modified to [9]: .
That representation demonstrates the modification, which appears in the second bracket. The whole term can be simplified to: It was pointed out that the formalism, here applied to a frequency dependent bulk viscosity, can also be used for a frequency dependent shear viscosity [2,37,38]. For monatomic liquids a transition from a viscous response to an elastic response of the shear viscosity occurs in the ps time regime at Maxwell's relaxation time τ M , the time range where neutron scattering and inelastic x-ray scattering observe density fluctuations. The response of a liquid to high-frequency density fluctuations can be described through a viscoelastic reaction of the fluid to a perturbation [3]. Maxwell's relaxation time τ M determines the reaction to a shear deformation of a volume element in the liquid. Below a frequency 1/τ M the liquid reacts viscous and beyond this frequency the liquid will respond in a solid-like manner. Formally this transition can be described by an exponentially decaying memory function which leads within the Zwanzig-Mori formalism to a transition in the dispersion relation from a low frequency sound velocity to a highfrequency sound velocity. The liquid is not able to relax the shear deformation within the frequency of the traveling sound mode and the mode propagates in a solid-like medium. This relaxation process is responsible for the modification of the LP ratio. The definition of the high-frequency sound velocity involves the potential of the liquid particles [3]. The highfrequency sound velocity is well-defined toward Q → 0 and hence the calculated modified LP-ratio will not approach the hydrodynamic value toward small wave vectors. The highfrequency sound velocity c ∞ (Q) has been calculated utilizing a classical potential for liquid rubidium [18,40]. The MDsimulation describes quite well the experimental spectra and dispersion relations and hence the high-frequency sound velocities can be regarded as quite reliable.
For the low-frequency sound velocity in this microscopic wave vector range we assume a wave vector dependent isothermal sound velocity c 0 (Q). It was argued that in the microscopic wave vector range the adiabatic sound propagation process is no more valid and turns into the isothermal for liquid metals [16]. It is conceivable that density fluctuations with wave lengths of a few atomic distances will occur in an isothermal environment and hence we have chosen the wave vector dependent isothermal sound velocity for the low frequency sound velocity. The isothermal sound velocity is related to the isothermal compressibility c 2 0 = 1 n m χT and through χ T with the structure factor [4]. The sound velocity can then be generalized to a Q-dependent sound propagation velocity: c 2 0 (Q) = kBT mS(Q) . With the experimental structure factor data S(Q) [39] c 0 (Q) can be calculated. Consequently we will consider the viscoelastic response of the liquid as the dominating relaxation process which is at the origin for a sound velocity change from a wave vector dependent isothermal c 0 (Q) to a high-frequency sound velocity c ∞ (Q): In figure 2 we plot the area ratios (circles) Ic 2IB obtained from the fit. Included is the expected hydrodynamic LP-ratio as a line. LP modified is calculated from equation (4) and plotted as stars. These ratios are around 0.8 and hence deviate strongly from the value of 0.1 from hydrodynamics. Toward larger Q values the ratio increases dramatically when the structure factor maximum, which is at Q = 1.5 Å −1 for rubidium, is approached. At these wave vectors the dominating quasielastic peak will increase the intensity ratio. In the future it would be interesting to assess this wave vector range whether LP modified is able to describe this wave vector range.
The calculated modified LP-ratios agree reasonably well with the intensity ratios. Note that LP modified is calculated from quantities independent from the measured experimental spectra. Around Q ≈ 0.45 Å −1 the intensity ratio provides values larger than LP modified . This behavior might be a hint that a further relaxation process is at work, at least in this wave vector range. That the dynamics of liquid metals may need a further relaxation process for a proper description of spectra was reported some time ago [16,33]. A further relaxation process improved, in particular, the modeling of the central line, however, it was a modest additional amount necessary with about 10% of the main relaxation process. The previous analysis of the liquid rubidium data did not need a further relaxation process [18]. A different explanation could be related to third order contamination from the spectrometer, which would have been at neutron energies of about 270 meV in the experiment [31].
At the smallest Q-vector the intensity ratio is reduced to about 0.4, indicating the transition to hydrodynamics within this wave vector regime. Nevertheless, the intensity ratio is still a factor 4 larger than expected from hydrodynamics. Careful analysis of the small wave vector dispersion relation in liquid cesium and lithium showed that in this wave vector range the experimental high-frequency sound velocity starts to deviate from c ∞ (Q) to smaller values and decreases toward the adiabatic value [14,16], but did not reach it. Despite these deviations the modified LP-ratio is able to describe the experimental data quite well and the most important relaxation process seems to be covered by this description.
In the past, wave vector dependent generalizations of the specific heats c P,V (Q) were investigated. It was reported that the specific heat ratio does not grow with increasing wave vector, rather decreases toward 1 with increasing wave vector [41,42]. Hence, the wave vector dependence of the specific heats can not be responsible for the deviation of the LP-ratio in the microscopic wave vector range. This behavior supports our approach to use the isothermal sound velocity in equation (4).

Conclusions
Neutron scattering spectra of liquid rubidium near the melting point have been analyzed with the aim to obtain the LP ratio at wave vectors beyond hydrodynamics. For this purpose the intensities of the central line and inelastic modes have been determined from the experiment. The derived intensity ratio is distinctly different from the prediction inferred from the fluid dynamics description based on hydrodynamic density fluctuation theory. The intensity ratio is about a factor 8 larger than predicted by hydrodynamics. This result points to an additional relaxation process, which is identified as the viscoelastic reaction of the liquid toward high frequencies. A modified LP ratio is able to describe the intensity ratio reasonably well. Since all the other liquid alkali metals have similar dynamic properties [20,43] we argue that the modified LP-ratio can be applied to all alkali metals. Other liquid metals have also quite small specific heat ratios and the spectra have been modeled with a similar formalism as for liquid rubidium. It needs to be investigated whether the proposed modification of the LP ratio is valid for liquid metals more generally. Furthermore, it would be interesting to investigate the case of non-metallic liquids, where the specific heat ratio is large compared to liquid metals.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.