High frequency dynamics in liquid nickel: an IXS study
S. Cazzato, T. Scopigno, S. Hosokawa, M. Inui, W-C. Pilgrim, G. Ruocco
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r High frequency dynamics in liquid nickel: an IXS study
S. Cazzato , , T. Scopigno , , S. Hosokawa , M. Inui , W-C. Pilgrim and G. Ruocco , Dipartimento di Fisica and INFM, Universit`a di Roma La Sapienza, I-00185 Roma, Italy. INFM CRS-SOFT, c/o Universit`a di Roma La Sapienza, I-00185, Roma, Italy. Department of Materials Science, Hiroshima University, Higashi-Hiroshima 739, Japan. Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739, Japan. and Institute of Physical Chemistry and Materials Science Centre,Philipps-University of Marburg, 35032 Marburg, Germany.
Owing to their large relatively thermal conductivity, peculiar, non-hydrodynamic features areexpected to characterize the acoustic-like excitations observed in liquid metals. We report here anexperimental study of collective modes in molten nickel, a case of exceptional geophysical interest forits relevance in Earth interior science. Our result shed light on previously reported contrasting evi-dences: in the explored energy-momentum region no deviation from the generalized hydrodynamicpicture describing non conductive fluids are observed. Implications for high frequency transportproperties in metallic fluids are discussed.
PACS numbers:
I. INTRODUCTION
Half a century of inelastic neutron scattering experi-ments, recently complemented by similar investigationswith X-rays, have clearly shown that acoustic like exci-tations can be sustained by simple liquids down to wave-lengths comparable to the mean interparticle distancesand frequencies extending up to the THz region.
Inordinary, non conductive fluids, such collective modescan be roughly described in terms of adiabatic soundwaves of energy E = ~ Ω propagating with sound ve-locity c s = lim Q → Q ) Q . This is a well known, imme-diate consequence of the hydrodynamic treatment whenthe condition Ω >> D T Q holds, where D T is the ther-mal diffusion coefficient for the system under study and Q the exchanged momentum between the probe and thesample. A deeper analysis reveals that such acousticwaves are actually subject to relaxation processes relatedto the frequency dependent viscosity. The origin of suchprocesses is twofold: the structural relaxation, respon-sible for dynamical arrest in those systems capable ofsupercooling, and a microscopic process which inducesan additional damping of the sound waves due to thenon-plane wave nature of the instantaneous vibrationaleigenmodes. Both these processes manifest themselvesat wavevectors Q ≈ / Q M , being Q M the principalmaximum of the static structure factor. This region, un-fortunately, represents the technical limit for both neu-tron and X-ray inelastic spectroscopy, hindering the com-plete understanding of the underlying dynamical pro-cesses which are still controversially debated. In liquid metals, an additional complication arises.The hydrodynamic condition Ω( Q ) >> D T Q breaksdown in the range 0 . . Q . − , an esti-mate obtained neglecting the Q dependence of the ther-mal conductivity. Consequently, an isothermal regimemay be expected to occur above the Ω( Q ) ≈ D T Q crossover, with the adiabatic limit attained only belowthis value. Among the implications of such an isother- mal regime on the sound waves propagation, noteworthyfeatures would be: • a reduced value of the sound velocity c t = c s / √ γ ,with γ = C P /C V , the ratio of constant pressure toconstant volume specific heats. • a different expression for the thermal contributionto the acoustic attenuation: ( γ − c t γD T instead of ( γ − D T Q .To address this issue, we present here an investigation ofthe high frequency dynamics in liquid nickel performedby Inelastic X-ray Scattering (IXS), a technique whichhas proved, since the early 90’s, to be an invaluable toolfor deepening our comprehension of the dynamics of sim-ple liquids on the microscopic scale, since it allows onone hand to overcome the kinematic limitations due tothe lower incident energies of the neutrons and their spe-cific energy-momentum relation, and on the other handprovides direct access to the coherent cross section of thescattering process. In this respect, previous investigations performed bymeans of Inelastic Neutron Scattering (INS) and molecu-lar dynamics (MD) have shown contrasting results.
In the former study the sound velocity attains indeedthe isothermal value at the longest accessible wavelength( Q ≈ − ), while in the latter the numerically es-timated sound velocity is always larger than the adia-batic value. Recently an interesting study involving bothquasi-elastic Neutron Scattering and MD has pointed outhow the simulation data for the speed of sound of liquidNi attain the adiabatic value c S at the lower accessed Q ’s( Q ≈ − ). Liquid Ni, indeed, is characterized by the largest spe-cific heat ratio among monatomic liquids, a propertywhich should emphasize any non hydrodynamic results.In addition, Ni is a system of paramount relevance ingeophysical science due to its presence in the Earth’sinterior. Indeed, the fact that the Earth outer core ismostly iron was established beyond reasonable doubt al-ready in the early 60’s, when it was confirmed that thedensity of the core was about 10% lower than the densityof iron, and that the seismic parameter Φ =
K/ρ , being K the bulk modulus and ρ the density, was higher thanthat of iron. About 4% Ni is thought to be present inthe core and although it does not appreciably change thedensity of liquid Fe, its presence should not be forgottenas phase diagrams of systems Fe-Ni-light elements maybe significantly different from those of systems withoutNi.
Among the most favorite candidates as lighter al-loying elements with Fe and Ni, sulfur is perhaps the mostaddressed one. In this respect, recently a high anoma-lous behavior of the ultrasonic sound velocity c S ( T ) andattenuation Γ( T ) as functions of temperature was re-ported in the mixture 85%Fe-5%Ni-10%S, for temper-atures above melting (T m =1650 K) up to 2000 K, atambient pressure conditions. In fact, and contrary tothe data of pure liquid metal components, the acousticvelocity is found to increase with temperature, as well asattenuation. A complete understanding of the underlyingbehavior of the alloy requires, in our opinion, to ascertainthe nature of acoustic excitations in pure liquid Ni.
II. THEORETICAL BACKGROUND:EXPECTED HYDRODYNAMIC BEHAVIOR FORLIQUID METALS
In an IXS experiment the double differential cross-section, which depends on the exchanged momentum Q and energy E , is proportional to the so called dynamicstructure factor S ( Q, ω ), which in turn is the Fouriertransform of the time dependent intermediate scatteringfunction F ( Q, t ) = 1 N X i,j D e − iQ · r i (0) e iQ · r j ( t ) E . (1)Here N is the total number of particles constituting thesystem, and r i ( t ) the position of particle i at time t . In particular, the zero time value of the intermediatescattering function is directly related to the structuralfeatures of the system, being F ( Q,
0) = S ( Q ), i.e. thestatic structure factor. In the hydrodynamic limit, forlow values of Q , the dynamic structure factor displaysthree distinct peaks. A quasi elastic one, located at zeroenergy exchange, whose width Γ e = D T Q is related tothe thermal diffusion coefficient D T , and two inelasticpeaks - the so called Brillouin doublet - located at fre-quencies ω = ± c S Q , with c S being the adiabatic speedof sound, and whose width depends mostly on kinematicviscosity, which is the main mechanism driving sounddamping. On the basis of linear hydrodynamics, the dis-persive behavior, i.e. the dependence of the frequency ω of propagating collective modes on Q , is well known todisplay a transition between a linear adiabatic regime toa linear isothermal one, characterized by an isothermalspeed of sound c T , at Q values such that ωτ th ∼
1, where τ th is the characteristic decay time of thermal fluctua-tions ( τ th ∼ /D T Q ). In other words, at sufficientlyhigh Q values, thermal fluctuations are expected to decayon a timescale much longer than the timescale of soundpropagation, which now takes place in a thermalized en-vironment. If we look at the lineshape of the dynamicstructure factor, the linewidth of the quasi-elastic linewill increase, ultimately overwhelming the Brillouin dou-blet and causing a shift of the inelastic peaks position.The adiabatic and isothermal speed of sound, as we al-ready pointed out in section I, are related by the specificheat ratio γ , and we may call Q ∼ c S /D T the value atwhich the adiabatic to isothermal transition takes place.In this scenario liquid metals constitute a particularlyinteresting class of simple liquids, because of their highthermal diffusivity with values of about ten times largerthan c S , shifting the Q -range of the isothermal regiondown to values between 0 . − and 3 nm − providedthat:1. the Q dependence of transport coefficients can beneglected below at least 3 nm − , as is the case en-countered in most liquid metals.2. the system interaction with radiation can be de-scribed within an effective single-component model,in which only the core electrons from the metal-lic ions interact with the electromagnetic radiation.However transport coefficients are supposed to takean effective value also carrying the net effect due tovalence electrons.The second assumption has been on the basis of a recentdebate focused on the interpretation of IXS data on liquidalkali metals and aluminum. The main difficultyin ascertaining the existence of sound propagation withlower than adiabatic sound velocity value lies in the factthat γ is for most metallic systems very close to unity,as can be seen from the specific heats ratios reported fora selection of liquid metals near their melting point intable I. For alkali metals γ ∼ III. THE EXPERIMENT
The experiment reported in this work was carriedout at the high resolution Beamline ID16 of the Euro-pean Synchrotron Radiation Facility (Grenoble, Fr). Thebackscattering monochromator and analyzer crystals, op-erating at the (11 , ,
11) silicon reflections gave a totalenergy resolution of 1.5 meV, while energy scans wereperformed by varying the temperature of the monochro-mator with respect to that of the analyzer crystals. Thesample, in the shape of a 100 µ m thick foil, was 99.993%purity nickel, purchased by Rare Metallic Co. Ltd. TABLE I: Thermal properties of selected liquid metals nearthe melting point. γ , the ratio of specific heats, and D T , thethermal diffusion coefficient, are reported at those temper-atures T, near the melting point, where data are available.For a more extensive summary of data on thermal as well asdynamical properties of liquid metals, see Ref. 3.Sample T [K] γ D T [nm / ps]Li 453 1.08 ,20.3 Na 371 1.12 Mg 923 1.29 Al 933 1.4 K 336.7 1.11 Fe 1808 1.8 Co 1765 1.8 Ni 1728 1.98 Cu 1356 1.33 Zn 693 1.25 Ga 303 1.08 Ge 1253 1.18
Rb 312 1.15 ,1.097 Ag 1233 1.32 Sn 505 1.11 Cs 302 44.6
308 1.102 , Au 1336 1.28 Hg 293 1.14 Pb 623 1.19 (Japan), and was hold in a sapphire cell obtained from asingle monocrystal. The use of sapphire prevented thesample interaction with container, still providing a goodcell transmission, with a total cell thickness of 500 µ mseen by the scattered beam. The cell was lodged in amolybdenum holder, and heated up by means of a prop-erly isolated tungsten resistance. The nickel absorptionlength at an energy of the incoming beam of 21 KeV, isabout 50 µ m, thus in our condition a 15% sample trans-mission was expected. A selection of IXS spectra fromliquid Ni at 1767 K is reported in Fig. 1 as open circlesat the lowest accessed Q values. Dotted lines report theexperimental resolution. Phonon modes from the sap-phire cell (speed of sound of about 11 kms − ) are wellrecognizable up to 3 nm − , however they are well sepa-rated from inelastic features from the sample at Q greaterthan 1 nm − . -30 -20 -10 0 10 20 30 0100200 E [meV]
Q = 2 nm -1 Q = 3 nm -1 -1 Q = 6 nm -1 Q = 5 nm -1 S ( Q , E ) [ u . a . ] Q = 8 nm -1 FIG. 1: (Color online) a : sound dispersion for liquid Ni:IXS ( (cid:3) ) from the present work; an INS investigation byBermejo et. al. ( N ); a MD simulation from Ruiz-Mart´ın etal. ( −·− ) . Adiabatic (—) and isothermal ( −−− ) sound dis-persions are also reported. b : Speed of sound reported fromthe present work ( (cid:3) ) and from INS ( N ) and MD ( − · − and ◦ ) investigations. The adiabatic, c S , and isothermal, c T , speed of sound are displayed ( · · · ). IV. RESULTS AND DISCUSSION
Experimental data have been analyzed with a dampedharmonic oscillator (DHO) function centered at fre-quency ω L , and of width Γ L modelling inelastic con-tributions from the metal. The elastic features havebeen represented by a Lorentzian of full width at halfmaximum (FWHM) Γ C . Such an approximation is ex-pected to work well for not too high Q ’s, i.e. until thereis a clear separation between elastic and inelastic fea-tures, as is the present case for IXS measurements. At Q values approaching Q M , corresponding to the staticstructure factor maximum, one has to resort to a modelbased on a so-called extended heat mode and two ex-tended sound modes, also incorporating the frequencysum rules. The elastic contribution from the cell, sup-posed to be much narrower than the experimental resolu-tion, has been modelled with a Dirac delta function cen-tered at zero frequency. Thus the model for the S ( Q, ω ) ba C T C S s ound s p ee d w L / Q [ k m / s ] s ound fr e qu e n c y w L [ m e V ] Q [ nm -1 ] FIG. 2: (Color online) a : sound dispersion for liquid Ni:IXS ( (cid:3) ) from the present work; an INS investigation byBermejo et. al. ( N ); a MD simulation from Ruiz-Mart´ın etal. ( −·− ) . Adiabatic (—) and isothermal ( −−− ) sound dis-persions are also reported. b : Speed of sound reported fromthe present work ( (cid:3) ) and from INS ( N ) and MD ( − · − and ◦ ) investigations. The adiabatic, c S , and isothermal, c T , speed of sound are displayed ( · · · ). can be written as I q ( ω ) = I C, ( Q ) δ ( ω ) + I C, ( Q ) Γ C ( Q ) ω + Γ C ( Q ) ++ I L ( Q ) ω L ( Q )Γ L ( Q )( ω − ω L ( Q )) + ( ω Γ L ( Q )) . (2)Additionally, in order to properly reproduce the experi-mental spectra, the above expression has to be modifiedsuch as to be compliant with the detailed balance condi-tion: S ( Q, ω ) = β ~ ω − e − β ~ ω I q ( ω ). (3)Before comparison with IXS data, the model is convo-luted with the experimental resolution. This procedureresults in the curves displayed as full lines in Fig. 1. Fig.2 reports the dispersion (a) and sound velocity (b) ob-tained for liquid Ni from the DHO model parameter ω L at different Q values ( (cid:3) ), while Fig. 3.a reports sound b d a m p i ng G L [ p s - ] Q [ nm -1 ] a e l a s ti c li n e w i d t h G C [ p s - ] FIG. 3: (Color online) a : sound damping from the presentwork ( (cid:3) ) and from INS ( N ). The dashed line ( − − − ) isa quadratic best fit to the IXS data. b : the quasielasticlinewidth obtained in the present investigation of liquid Ni( (cid:3) ) and from a recent INS investigation (from the coherentdynamic structure factor) ( (cid:7) ). The expected hydrodynamicbehavior is also reported ( − − − ), from the knowledge of thethermal diffusion coefficient D T . damping as derived from the DHO parameter Γ L ( (cid:3) ).Sound speed and damping data are also reported fromthe cited INS experiment ( N ) , as well as from MD ex-periments ( ◦ , − · − ). On one hand our results ( (cid:3) ) forthe acoustic excitations frequency dependence on Q arein very good agreement with the data obtained by Ruiz-Mart´ın et al. ( − · − ) in the most recent MD simulationon the subject, while on the other hand the experimen-tal INS data ( N ) are qualitatively different from bothsets of data along the momentum range investigated. This discrepancy is particularly evident in the region of
Q <
12 nm − , where INS estimated sound speed reachesthe isothermal value c T (Fig. 2). This could be ascribedto the kinematic limitations to the energy accessible win-dow at the lowest Q ’s which prevented the observation ofthe S ( Q, ω ) tail, and thus a reliable estimate of the DHOmodel parameters. In order to address the last issue, in Fig. 4 we report acomparison of IXS ( ◦ ) and INS ( N , ) dynamic structurefactors at Q = 8 nm − , corresponding to the lowestaccessed Q -point for neutrons. The inelastic contribu-tion to the IXS spectrum is emphasized by reportingthe DHO function derived from the model (Eqs. (2)and (3)). Beside the quasielastic peak, reflecting alsothe effect of incoherent scattering (which is not presentin IXS in the case of monatomic systems), INS datashow on the anti-Stokes side the presence of an inelasticpeak around -16 meV. However, although the detailedbalance condition would imply an even more pronouncedpeak on the Stokes side, its symmetric counterpartcannot be observed. Last but not least, the energy of -40 -20 0 20 4010100 S ( Q , E ) [ u . a . ] E [meV]
FIG. 4: Comparison of IXS ( ◦ ) and INS ( N ) dynamic struc-ture factors for liquid Ni at Q = 8 nm − . Data have beennormalized to a constant factor in order to plot them on thesame scale. The DHO function used to model IXS inelasticfeatures is also reported (—, see also Fig. 1). The exchangedenergy range covered by INS is narrower than that accessedby IXS, due to the kinematics of the scattering process forneutrons. propagating excitations found by means of IXS (22.6meV) lies just at the edge of the accessed energy rangefor INS (-20 to 20 meV; see Fig. 4), corresponding tothe energy and scattering angle configurations used inthis experiment. Summing up, while at sufficiently high Q ’s the sound dispersion of liquid Ni lies well inside the region of the momentum-energy range covered byINS, at Q values approaching wavevectors as low as8 nm − the reliability of the dispersion curve extractedby INS is questionable. In Fig. 3.b, we report datafor the elastic linewidth, compared with the expectedhydrodynamic behavior, in which case the width of thequasielastic line is entirely due to thermal relaxationΓ C = D T Q ( − − − ). Experimental INS values for Γ C are also shown from a recent work ( (cid:7) ). The presentset of data (Fig. 3.b) seems to unsupport the hypothesisof a weak Q dependence of transport coefficients. Under this circumstance, the crossover to an isother-mal regime in liquid Ni may occur at values well below1 ÷ − , thus not accessible in the present experiment. V. CONCLUSIONS
In conclusion, we performed an IXS experiment on liq-uid Ni at 1767 K, showing the ability of such systemto sustain sound propagation over wavelengths compa-rable to the typical interparticle distance. Despite theindication from a previous INS investigation suggest-ing the tendency of sound speed to attain the isother-mal value at Q lower than 8 nm − , we found the evi-dence of an adiabatic dynamical regime holding at Q aslow as 2 nm − , thus confirming MD results for the mi-croscopic dynamics of this system. The discrepancy,observed for 8 < Q <
12 nm − between INS data onone hand and results from IXS and MD on the otherhand, is probably related to the limitations imposed tothe energy-momentum accessible region by the kinemat-ics of the scattering process for neutrons. Even though anadiabatic regime still holds at the lowest accessed Q forliquid Ni, the weak Q dependence of the quasielastic line-width clearly suggests that a generalized hydrodynamicpicture should be invoked. The present result calls forfurther investigations with higher signal to noise ratioand at different temperatures to unambiguously ascer-tain the existence of an intermediate isothermal regimein liquid Ni in the explored momentum region. J.-P. Hansen and I. McDonald,
Theory of simple liquids (Academic, New York, 1986). U. Balucani and M. Zoppi,
Dynamics of the liquid state (Clarendon Press, Oxford, 1983). T. Scopigno, G. Ruocco, and F. Sette, Rev. Mod. Phys. , 881 (2005). T. Faber,
Introduction to the Theory of Liquid Metals (Cambridge University Press, Cambridge, 1972). T. Scopigno and G. Ruocco, JNCS , 3160 (2007). F. J. Bermejo, M. L. Saboungi, D. L. Price, M. Alvarez,B. Roessli, C. Cabrillo, and A. Ivanov, Phys. Rev. Lett. , 106 (2000). M. M. G. Alemany, C. Rey, and L. J. Gallego, Phys. Rev. B , 685 (1998). G. Alemany, O. Dieguez, C. Rey, and L. J. Gallego, Phys.Rev. B , 9208 (1999). F. J. Cherne, I. M. Baskes, and P. A. Deymier, Phys. Rev.E , 024209 (2001). M. D. Ruiz-Mart´ın, M. Jim´enez-Ruiz, M. Plazanet, F. J.B. F. J, R. Fern´andez-Perea, and C. Cabrillo, Phys. Rev.B , 224202 (2007). J.-P. Poirier, Phys. Earth Planet. Inter. , 319 (1994). F. Birch, J. Geophys. Res. p. 4377 (1964). R. Brett, Geochim. Cosmochim. Acta , 203 (1971). P. Waldner and A. D. Pelton, Metall. Mater. Trans. ,897 (2004). P. M. Nash, M. H. Manghnani, and R. A. Secco, Science , 219 (1997). T. Iida and R. I. L. Guthrie,
The Physical Properties ofLiquid Metals (Oxford Science Publications, 1993). R. Hultgren, P. Desai, D. T. Hawkins, M. Gleiser, K. K.Kelly, and D. D. Wagman,
Selected Values of the Ther-modinamic Properties of the Elements (American Societyfor Metals, 1973). R. Ohse,
Handbook of Thermodynamic and TransportProperties of Alkali Metals (Blackwell Scientific Publica-tions, 1985). Y. S. Touioukiam and C. Y. Ho,