Microscopic dynamics in liquid metals: The experimental point of view

The experimental results relevant for the understanding of the microscopic dynamics in liquid metals are reviewed, with special regard to the ones achieved in the last two decades. Inelastic neutron scattering played a major role since the development of neutron facilities in the 1960s. The last ten years, however, saw the development of third generation radiation sources, which opened the possibility of performing inelastic scattering with x rays, thus disclosing previously unaccessible energy-momentum regions. The purely coherent response of x rays, moreover, combined with the mixed coherent or incoherent response typical of neutron scattering, provides enormous potentialities to disentangle aspects related to the collectivity of motion from the single-particle dynamics. If the last $20\text{years}$ saw major experimental developments, on the theoretical side fresh ideas came up to the side of the most traditional and established theories. Beside the raw experimental results therefore models and theoretical approaches are reviewed for the description of microscopic dynamics over different length scales, from the hydrodynamic region down to the single-particle regime, walking the perilous and sometimes uncharted path of the generalized hydrodynamics extension. Approaches peculiar of conductive systems, based on the ionic plasma theory, are also considered, as well as kinetic and mode coupling theory applied to hard-sphere systems, which turn out to mimic with remarkable detail the atomic dynamics of liquid metals. Finally, cutting edge issues and open problems, such as the ultimate origin of the anomalous acoustic dispersion or the relevance of transport properties of a conductive system in ruling the ionic dynamic structure factor, are discussed.

Figure 1

Ratio between the total number of photons scattered by the Thomson process and those lost through all the other processes, among which is photoelectric absorption, in a sample of length ${\mu }^{-1}$ calculated as a function of the atomic number $Z$ for photons of incident energy of $\approx 22\mathrm{keV}$.

Figure 2

Kinematic region accessible to IXS in reduced $E∕E_i$ and $Q∕k_i$ units (left panel). The right panel shows the realistic cases of molten lithium (open circles) and sodium (full circles) for an incident energy of $25\mathrm{keV}$.

Figure 3

Kinematic region accessible to neutron-scattering experiments for incident energies $E_i=50\mathrm{meV}$ (left panel) and $E_i=500\mathrm{meV}$ (right panel), reported for different scattering angles. Open and full circles are the (approximated) sound dispersion of molten lithium and sodium, respectively.

Figure 4

The coherent scattering cross section of the elements for x ray (open circles) and for neutrons (full circles) reported as a function of the atomic number $Z$.

Figure 5

Coupling IXS with levitation techniques: constant $Q$ spectra of liquid alumina. From 177.

Figure 6

Pioneering (1991) low-resolution IXS determination of the dynamic structure factor in liquid lithium with INELAX. From 37.

Figure 7

Dispersion curve of liquid lithium achieved with IXS at INELAX. Theoretical predictions and INS results at higher $Q$’s are also reported. From 37.

Figure 8

First IXS measurements on liquid lithium performed on a third generation source (ESRF). Energy resolution is $\delta E=11\mathrm{meV}$. The continuous line is the best fit according to the extended hydrodynamic model of Eq. (63). From 178.

Figure 9

IXS measurement of liquid lithium in a wide energy-momentum region. The transition from hydrodynamic to Gaussian-like response (continuous line in the right panel) can be clearly noticed. Energy resolutions are 1.5, 3.0, and $7.0\mathrm{meV}$, increasing with the exchanged momentum. From 159.

Figure 10

Sound velocity as deduced by the maxima of the current correlation spectra, from the best fit with quantum corrected and resolution convoluted models.

Figure 11

Memory function at work: refined line-shape analysis of high-resolution IXS spectra. Both thermal and viscous channel are taken into account, mimicking this latter with single [Eq. (56), dotted line] or two [Eq. (59), continuous line] exponential processes.

Figure 12

Constant energy slices of the dynamic structure factor determined by IXS. Umklapp modes are visible on the sides of the main structure factor peak.

Figure 13

Randolph’s measurement (151) of the dynamic structure factor of liquid sodium in reduced units, $\alpha ={\hslash }^2{Q}^2∕2mT$ and $\beta =\hslash \omega ∕T$, for four different values of momentum transfer. Line-shape analysis according to different models (69) Continuous line: $\mathrm{EMA}+\text{calculated}$ MSD. Short dashed line: $\mathrm{EMA}+\text{computer}$ simulation computation of the MSD. Long dashed line: hydrodynamic prediction. Dot-dashed line: free streaming limit.

Figure 14

Reduced quasielastic linewidth ${\omega }_{1∕2}∕D{Q}^2$ in liquid sodium at three different temperatures (136). The dotted line is the Fickian limit while the dash-dotted line is the perfect gas behavior $(\propto 1∕Q)$. The dashed line is the hard-sphere prediction (56), while the continuous line is the result obtained with mode coupling theory (92).

Figure 15

Left panel: IXS determination of the $S(Q,\omega )$ in liquid sodium for selected temperatures. Right panel: dispersion curves at different temperature.

Figure 16

Sketch of the kinematic regions accessed in the experiments of Bove et al. (dashed line) and Cabrillo et al. (continuous lines). Linear sound dispersion is also reported.

Figure 17

Quasielastic linewidth according to recent INS measurements. Full and open circles are the coherent and incoherent contributions, respectively, determined with time of flight (41). Open triangles are the incoherent linewidth as measured with triple axis spectrometer (25), the observed $2\text{−}\mathrm{meV}$ coherent contribution is also indicated. The lower dotted line indicates the Fickian approximation.

Figure 18

Left panel: time-of-flight determination of the DSF in liquid potassium (circles) (41). The dash-dotted line depicts the coherent contribution. Right panel: triple axis spectrometer measurements in a similar momentum transfer region (circles) (25). The continuous line is the inelastic contribution to the collective dynamics.

Figure 19

High-resolution IXS measurements in liquid potassium (open circles) (132). The continuous line is the line-shape description according to a multiple relaxation memory function model (see text).

Figure 20

Viscous relaxation times as measured by means of IXS (132). Circles: structural relaxation time. Triangles: microscopic relaxation time. The relaxation time obtained by means of INS is also reported (41), showing how it averages between the two mechanisms reported by IXS.

Figure 21

Dispersion curves (maxima of the current correlation function) measured by INS [open circles (25), stars (41)] and IXS [full circles (132)].

Figure 22

Sound velocities as determined by IXS: apparent [circles, from the maxima of $C_L(Q,\omega )$], isothermal (dotted line), and infinite frequency limit (continuous line) determined from Eqs. (46, 52). The unrelaxed sound velocity values, due to the structural relaxation only (down triangle) and to the whole relaxation process (uptriangle), are also reported as estimated by the fitting.

Figure 23

The experimental determination of constant $Q$ slices of the DSF by means of INS scattering in liquid rubidium. A new era for the study of collective properties in simple liquids. From 59.

Figure 24

Dynamic structure factor of cesium at the melting point (circles). The continuous line is the viscoelastic approximation. From 22.

Figure 25

Linewidth of the coherent quasielastic spectral component in molten cesium (circles). The continuous line is the thermal value $D_T{Q}^2$, while the dashed line is the hard-sphere prediction from Eq. (67). From 22.

Figure 26

Dynamic structure factor of molten aluminum. Comparison between the OF-AIMD calculation (continuous line) (90) and the experimental IXS values with their fitting based on generalized hydrodynamics (162). From 89.

Figure 27

Sound dispersion of liquid aluminum from the maxima of the current correlation function: open circles, OF-AIMD calculation (90); full circles, IXS experimental values (162). Dispersion from the maxima of the dynamic structure factor numerically evaluated is also reported (open triangles), as well as the hydrodynamic value (continuous line). From 90.

Figure 28

Resolution deconvoluted, classical line shape utilized to described the IXS spectra of molten Li (161, 162; 163), Na, and Al, reported on relative momentum and energy scale (see text).

Figure 29

INS constant energy scans (circles) compared to the model of Buchenau for plane-wave excitations in solids (35). From 18.

Figure 30

(Color online) Comparison between the INS [stars (18) and triangles (26)] and IXS [open circles with error bars (167)] determinations of $S(Q,\omega )$ in gallium at the melting temperature, for two different values of the (constant) momentum transfer. The dotted line is the viscoelastic prediction (18) (convoluted with the INS instrumental resolution and accounting for the detailed balance condition), while the continuous line is the best fit to the IXS data utilizing a memory function accounting for the thermal relaxation and two viscous processes (see text). Molecular-dynamics simulations for the coherent (dotted line) and total (dash-dotted line) $S(Q,\omega )$, convoluted to the INS resolution, are also reported (26).

Figure 31

(Color online) Half width at half maximum of the dynamic structure factor as determined by IXS in the kinetic regime (open circles). The prediction according to Enskog’s theory is also shown (continuous line) (165).

Figure 32

Sound velocity and attenuation in molten silicon (102).

Figure 33

FWHM of molten Bi compared to other simple liquids. It can be noted the presence of an intermediate minimum between the first two maxima which is not present in the other simple fluids. From 61.

Figure 34

DSF of molten Hg measured by means of INS at the indicated $Q$ values. The inelastic and quasielastic components, modeled with two Lorentzians and a DHO, respectively, are also shown. From 27.

Figure 35

Left panel: IXS determination of the DSF in Hg near the melting temperature. Right upper panel: dispersion relation and sound attenuation properties as deduced by a $\mathrm{DHO}+\text{one}$ Lorentzian fit. The low- and high-frequency limits are also reported. Right lower panel: Corresponding sound velocities. From 103.

Figure 36

Sound velocity in liquid nickel determined by INS (full circles) and MD (open circles). The isothermal $\left(c_t\right)$ and adiabatic $\left(c_s\right)$ values are also indicated.

Figure 37

(Color online) Random-phase approximation at work: data taken from Table and 26 for pure elements and from 86 and 157 for alkali-ammonia solutions, with the BS values estimated through Eq. (75). Full circles: the measured sound velocity $c$ is the maximum high-frequency value determined over the $Q$ range probed by inelastic-scattering techniques, i.e., includes the positive dispersion effect. Open circles: $c$ is the adiabatic value. The subscript of Li and Na indicates the concentrations in ammonia solutions. The Na group is relative to several different temperatures.

Figure 38

Sketch of the different dynamical regimes on decreasing the wavelength.

Figure 39

(Color online) Generalized longitudinal viscosities for a collection of liquid metals, extracted by the experimental data throughout a memory function approach. The long-wavelength limit, determined by ultrasonic measurements, is also reported for K, Na, and Ga (bottom to top).

Figure 40

Sound dispersion in several kind of liquids. Alkali metals [Li (159) and K (132), dashed line is the isothermal sound velocity]; hydrogen bonding systems [water (133) and hydrogen fluoride (7)]; low- and high-frequency sound velocities are also indicated by the dotted lines; liquid neon (60); adiabatic (circles and line) apparent (open triangles) and high-frequency (open diamonds) sound velocities; liquid silica, molecular-dynamics simulations (98), in this case also the transverse branch is reported.

Figure 41

Sound velocities in several glassy systems. Hard-sphere glass, exact solution within MCT (91); metallic glass obtained quenching a model system interacting via Price-Tosi pseudopotential (170), the crystalline counterpart is also reported; Lennard-Jones glass obtained in a similar way (153), reported with the low- and high-frequency sound velocities; $\mathrm{Si}{\mathrm{O}}_2$, MD as in Fig. 40 but in the glassy state (98) and experimentally determined by means of IXS scattering (155). In both cases the transverse branch is also reported; glassy selenium (166), again the sound velocity exceeds the adiabatic value.