Unified description of sound velocities in strongly coupled Yukawa systems of different spatial dimensionality
UUnified description of sound velocities in strongly coupled Yukawa systems ofdifferent spatial dimensionality
Sergey A. Khrapak a) Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrumf¨ur Luft- und Raumfahrt (DLR), 82234 Weßling, Germany;Joint Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow,Russia (Dated: 17 October 2019)
Sound velocities in classical single-component fluids with Yukawa (screened Coulomb) interactions are sys-tematically evaluated and analyzed in one-, two-, and three spatial dimensions ( D = 1 , , (cid:112) Q / ∆ m , where Q is the particle charge, m is the particle mass, n is the particle density, and ∆ = n − / D is the unified interparticle distance. Thesound velocity can be expressed as a product of this scaling factor and a dimension-dependent function ofthe screening parameter, κ = ∆ /λ , where λ is the screening length. A unified approach is used to deriveexplicit expressions for these dimension-dependent functions in the weakly screened regime ( κ (cid:46) κ (cid:38) I. INTRODUCTION
Investigations into linear and non-linear waves in com-plex (dusty) plasmas – systems of charged macroscopicparticles immersed in a plasma environment – is an activeresearch area with many interesting topics, such as e.g.sound (dust-acoustic) waves, instabilities, Mach cones,shocks, solitons, and turbulence.
In experiments, suf-ficiently long wavelengths are usually easy accessible forinvestigation, which exceed considerably the characteris-tic interparticle separation. At these wavelengths collec-tive excitations exhibit acoustic-like dispersion and thesound velocities play central role in characterizing thesystem.The particle charge in complex plasmas is typicallyvery high (10 − elementary charges for a micron-range sized particles). Due to strong electrical repulsionbetween the particles they usually form condensed liq-uid and solid phases. It is well understood that disper-sion properties of strongly coupled complex plasmas sig-nificantly deviate from those characteristic for an idealgaseous plasma. Strong coupling effects affect themagnitudes of sound velocities.
Strongly coupled com-plex plasma fluids in two and three dimensions can sup-port transverse excitations at finite (sufficiently short)wavelengths.
Instability thresholds (e.g. of the ioncurrent instability) are shifted at strong coupling. Waves in complex plasmas are investigated in one-dimensinal (1D), two-dimensional (2D), and three-dimensional configurations (3D). 1D linear particle ar- a) Electronic mail: [email protected] rangements as well as 1D and quasi-1D particle rings areformed by creating appropriate confining potential con-figurations above the negatively charged surface (elec-trode), responsible for particle levitation.
2D andquasi-2D layers are extensively studied in laboratory ex-periments with radio-frequency (rf) discharges, where thelevitating particles form horizontal layer(s) in the plasmasheath above the lower rf electrode.
Waves in large3D particle clouds have been initially observed in a Q-machine, and then in dusty plasmas formed in a posi-tive column (sometimes stratified) of direct-current glowdischarges, as well as in various experiments undermicrogravity conditions. Sound velocities can be relatively easy and accuratelymeasured in experiments and contain importantinformation about the systems investigated.The purpose of this paper is to provide a unified de-scription of sound velocities in strongly coupled complexplasmas in 1D, 2D, and 3D geometries. It is assumed thatthe particles are interacting via the isotropic pairwiseYukawa (screened Coulomb) potential. Simple practicalformulas are obtained, which are applicable to condensedfluid and solid phases. In particular, it is demonstratedthat the sound velocities are given by the product of therelevant velocity scale (cid:112) Q / ∆ m and the screening func-tion f ( κ ), where Q is the particle charge, ∆ = n − / D isthe characteristic interparticle separation, n is the den-sity, D is the dimensionality, m is the particle mass, and κ is the screening parameter defined as the ratio of theinterparticle separation to the screening length λ , thatis κ = ∆ /λ . The properties of f ( κ ) in 1D, 2D, and 3Dcases are investigated. In particular, the two regimesof weakly screened ( κ (cid:28)
1) and strongly screened in- a r X i v : . [ phy s i c s . p l a s m - ph ] O c t teractions ( κ (cid:29)
1) are considered in detail. Importantconsequences and relations are discussed.Yukawa systems are characterized by the repulsive in-teraction potential of the form φ ( r ) = ( Q /r ) exp( − r/λ ).Regardless of dimensionality, the phase state of the sys-tem is conventionally described by the two dimensionlessparameters, which are the (Coulomb) coupling parame-ter Γ = Q / ∆ T , and the screening parameter κ , where T is the system temperature (in energy units, so that k B = 1). It is important to note that very often theWigner-Seitz radius is used as a length unit, instead of∆. The Wigner-Seitz radius is defined from 4 πna / πa n = 1 in 2D, and na = 1 in 1D (that is only in1D we have ∆ = a ). Correspondingly, Γ and κ are alsooften defined in terms of a and one should pay attentionto this. In this paper ∆ is exclusively used as a lengthunit.The Yukawa potential is considered as a reasonablestarting point to model interactions in complex (dusty)plasmas and colloidal dispersions, although in manycases the actual interactions (in particular, their long-range asymptotes) are much more complex. Thisis particularly true in cases when electric fields andion drifts are present, resulting in plasma wakes andwake-mediated interactions.
The sound velocitieswill be certainly affected by deviations from the assumedYukawa potential, but we do not attempt to discuss thisissue here. Recently, the effect of long-range deviationsfrom the pure Yukawa potential on the dispersion rela-tions of the longitudinal waves in isotropic complex plas-mas have been investigated. The behavior of waves ina 1D dusty plasma lattice where the dust particles inter-act via Yukawa plus electric dipole interactions has beentheoretically studied in Refs. 51 and 52.The paper is organized as follows. In section II theunified approach to the calculation of sound velocities instrongly coupled Yukawa systems in 1D, 2D, and 3D ispresented. Main results are summarized in Section III.Here the weakly screened regime is analyzed in detail.Approximate expressions for the sound velocities in sys-tems with steeply repulsive potentials are derived, andit is explained why spatial dimensionality does not affectconsiderably the magnitude of sound velocities in thisregime. This is followed by conclusion in Sec. IV. Rela-tion to a simple freezing indicator of classical 3D fluidsproposed earlier is then briefly discussed in Appendix .
II. SOUND VELOCITIES IN DIFFERENT SPATIALDIMENSIONS
Strongly coupled Yukawa systems support one longi-tudinal mode in 1D case, one longitudinal and one trans-verse mode in 2D case, and one longitudinal and twotransverse modes in 3D case.The longitudinal sound velocities can be obtainedfrom the conventional hydrodynamic (fluid) approach. This requires knowledge of an appropriate equation of state. The standard adiabatic sound velocity is c s = (cid:112) (1 /m )( ∂P/∂n ) s , where P is the pressure of a singlecomponent Yukawa system and the subscript s denotesthat the derivative with respect to density is taken atconstant entropy. Note that ( ∂P/∂n ) s = γ ( ∂P/∂n ) T ,where γ = c p /c v is the adiabatic index. For strongly cou-pled Yukawa systems we have γ (cid:39)
1, which is a generalproperty of soft repulsive interactions.
This fluid ap-proach has been exploited previously for Yukawa systemsin 3D case as well as in 2D case. Generalization to1D case is trivial.The sound velocities of strongly coupled Yukawa sys-tems can also be obtained from infinite-frequency (in-stantaneous) elastic moduli, directly related to the in-stantaneous normal modes.
This approach is appli-cable to fluids and solids and allows to calculate both thelongitudinal and transverse sound velocities in a univer-sal manner and hence is adopted here.The elastic waves modes (instantaneous normalmodes) in the strongly coupled plasma-related fluidsare rather well described by the quasilocalized chargeapproximation (QLCA), also known as the quasi-crystalline approximation (QCA).
This approxi-mation relates wave dispersion relations to the interparti-cle interaction potential φ ( r ) and the equilibrium radialdistribution function (RDF) g ( r ), characterizing struc-tural properties of the system. It can be considered as ei-ther a generalization of the random phase approximationor as a generalization of the phonon theory of solids. The latter point of view is particularly relevant, becausein the special case of a cold crystalline solid the QCAdispersion reduces to the ordinary phonon dispersion re-lation, justifying the approach name. It is known thatfor 2D Yukawa systems, the angularly averaged latticedispersions are remarkably similar to the isotropic QCAfluid dispersions. It is not very unreasonable to ex-pect similar behavior in the 3D case.The long-wavelength limits of the QCA dispersion re-lations can be used to define the elastic longitudinal andtransverse sound velocities, c l and c t , as explained in de-tail below. The relation to the thermodynamic (adiabatic (cid:39) isthermal) sound velocity is then c s (cid:39) c l − (4 / c t in3D and c s = c l − c t in 2D. For Yukawa interactions(as well as for other soft long-ranged repulsive inter-action potentials) the strong inequality c l (cid:29) c t holdsat strong coupling. This implies that we have approx-imately c s (cid:39) c l . The accuracy of this relation hasbeen numerously tested for strongly coupled Yukawa flu-ids, as well as other soft interactions, bothin 3D and 2D cases.The general QCA (QLCA) expressions for the longitu-dinal and transverse dispersion relations are ω l = nm (cid:90) ∂ φ ( r ) ∂z g ( r ) [1 − cos( kz )] d r , (1)and ω t = nm (cid:90) ∂ φ ( r ) ∂x g ( r ) [1 − cos( kz )] d r , (2)where ω is the frequency and k is the wave vector. It isworth mentioning at this point that ω l and ω t can beidentified as the potential (excess) contributions to thenormalized second frequency moments of the longitudi-nal and transverse current spectra, C l/t ( k, ω ). Kineticterms, which are absent in the QCA approach [3(
T /m ) k for the longitudinal branch and ( T /m ) k for the trans-verse one], are relatively small at strong coupling. Thus,the formal essence of the QCA approach is just to ap-proximate the actual dispersion relations by the excesscontributions to the second frequency moments of thecorresponding current spectra.We proceed further as follows. The derivatives of thepair interaction potential in Eqs. (1) and (2) are evalu-ated from ∂ φ ( r ) ∂x α = φ (cid:48)(cid:48) ( r ) x α r + φ ( r ) (cid:48) r (cid:18) − x α r (cid:19) , where x α = x, y, z in 3D, x α = x, z in 2D, x α = z in 1D,and r = (cid:112)(cid:80) α x α . Note also that from symmetry ∂ φ ( r ) ∂x = ∂ φ ( r ) ∂y = 12 (cid:20) ∆ φ ( r ) − ∂ φ ( r ) ∂z (cid:21) in 3D, and ∂ φ ( r ) ∂x = ∆ φ ( r ) − ∂ φ ( r ) ∂z in 2D.Let us consider isotropic fluids with pairwise interac-tions of the form φ ( r ) = (cid:15)f ( r/σ ) , (3)where (cid:15) is the energy scale and σ is the length scale.Except for some special cases (in the present contextthis corresponds to the unscreened Coulomb interactionlimit, which will not be considered explicitly), the long-wavelength dispersion is acoustic:lim k → ω l k = c l , lim k → ω t k = c t . (4)The emerging elastic longitudinal and transverse soundvelocities can be presented in a universal form c l/t = ω D σ (cid:90) ∞ dxx D +1 g ( x ) (cid:20) A f (cid:48) ( x ) x + B f (cid:48)(cid:48) ( x ) (cid:21) , (5)where x = r/σ is the reduced distance. The D -dimensional effective frequencies ω D and the coefficients A and B are summarized in Table I. The last line in Ta-ble I simply reflects the fact that the transverse mode isabsent in 1D case and the integration over the positiveand negative parts of z -axis is equivalent to the doubledintegration over the positive part.An important remark about the transverse dispersionrelation in fluids should be made at this point. Althoughstrongly coupled (dense) fluids do support the transverse TABLE I. The coefficients A l/t and B l/t appearing in Eq. (5)for the longitudinal ( l ) and transverse ( t ) sound velocities, aswell as D -dimensional nominal frequencies and the coefficients C D in 3D, 2D, and 1D spatial dimensions. D ω D C D A l B l A t B t
3D 4 πn(cid:15)σ/m π
115 110 215 130
2D 2 π(cid:15)n/m π
116 316 316 116
1D 2 (cid:15)n/mσ waves propagation, their dispersion is somewhat differentfrom that in a solid. The existence of transverse modesin fluids is a consequence of the fact that their responseto high-frequency short-wavelength perturbations is sim-ilar to that of a solid. However, shear waves in fluidscannot exist for arbitrary long wavelengths. The min-imum threshold wave number k ∗ emerges, below whichtransverse waves cannot propagate. This phenomenon,often referred to as the k -gap in the transverse mode, isa very well known property of the fluid state. Lo-cating k ∗ for various simple fluids in different parameterregimes and investigating k -gap consequences on the liq-uid state properties is an active area of research. Forour present purpose it is important that the inclination ofthe dispersion curve ∂ω t /∂k near the onset of the trans-verse mode at k > k ∗ can be well approximated by c t .Thus, the latter is a meaningful quantity both in solidand strongly coupled fluid states.Next we take σ = ∆ and assume Yukawa interactionpotential between the particles. This implies (cid:15) = Q / ∆and f ( x ) = exp( − κx ) /x . The expressions for the longi-tudinal and transverse sound velocities become c l/t = C D (cid:18) Q ∆ m (cid:19) (cid:90) ∞ dxx D− exp( − κx ) g ( x ) (cid:2) B l/t κ x + (2 B l/t − A l/t )(1 + κx ) (cid:3) . (6)The numerical coefficients C D are provided in Table I.At this point it is also useful to introduce the universalvelocity scale c = (cid:112) Q / ∆ m . Note that c = √ Γ v T ,where v T = (cid:112) T /m is the thermal velocity.The excess internal (potential) energy can also be ex-pressed using the RDF and the pair interaction potential.The expression for the excess energy per particle in unitsof temperature is u ex = n T (cid:90) d r φ ( r ) g ( r ) . (7)For the Yukawa interaction potential in D dimensionsthis yields u ex = C D Γ2 (cid:90) ∞ dxx D− exp( − κx ) g ( x ) , (8)where we have used the identity (cid:15)/T = Q / ∆ T ≡ Γ.Finally, the following line of arguments is used. In thespecial case of a cold crytalline solid, the RDF representsa series of delta-correlated peaks corresponding to a givenlattice structure. Assuming that the lattice structure isfixed (in fact, the equilibrium lattice structure changesfrom bcc to fcc when κ increases in 3D case, but thisis not important for our present purpose) the RDF is auniversal function of x : g ( x ; Γ , κ ) = g ( x ) (for simplicitywe keep isotropic notation). Independence of g ( x ) of κ allows us make use of the following identities: C D Γ (cid:90) ∞ dxx D− exp( − κx ) g ( x ) = 2 u ex , C D Γ (cid:90) ∞ dxx D− κx exp( − κx ) g ( x ) = − κ ∂u ex ∂κ , C D Γ (cid:90) ∞ dxx D− κ x exp( − κx ) g ( x ) = 2 κ ∂ u ex ∂κ . These expressions are exact for crystalline lattices, butremain good approximations in the strongly coupled fluidregime. In particular, the dependence g ( x ; Γ , κ ) on κ isknown to be very weak for weakly screened ( κ is not muchlarger than unity) Yukawa fluids. The excess energyat strong coupling can be very accurately approximatedas u ex (cid:39) M fl Γ (cid:39) M cr Γ, where M fl and M cr can be re-ferred to as the fluid and crystalline Madelung constants( M fl ∼ M cr ). This reflects the fact that for soft repul-sive interactions the dominant contribution to the excessenergy comes from static correlations. One can under-stand this as follows. For soft long-ranged interactionsthe integral in Eq. (7) is dominated by long distances,where g ( x ) exhibits relatively small oscillations aroundunity (for finite temperatures). The ratio u ex / Γ is thennot very sensitive to the exact shape of g ( x ) at small x (provided the correlation hole radius is properly ac-counted for) and, hence, to the phase state of the system.The consideration above implies that if u ex (and itsdependence on κ ) is known, the integrals appearing in theexpressions for sound velocities can be evaluated. Belowwe demonstrate how this works in practice in 1D, 2D,and 3D cases. A. 1D case
The excess energy of an equidistant chain of particlesis u ex = Γ ∞ (cid:88) j =1 e − κj j = Γ [ κ − ln( e κ − . (9)After simple algebra we get c l = c (cid:26) κe κ [ κ − e κ ]( e κ − − e κ − (cid:27) . (10) Δ / λ c l / c FIG. 1. Reduced longitudinal sound velocities versus thescreening parameter κ = ∆ /λ for Yukawa systems in differentspatial dimensions. The three solid curves from top to bot-tom correspond to 3D, 2D, and 1D cases, respectively. Thedashed curve corresponds to the conventional DLW scale ofEq. (12). This result has been previously reported in Ref. 86. Itcan be also obtained by direct summation c l = c (cid:90) ∞ dxg ( x ) e − κx (2 + 2 κx + κ x ) /x = c ∞ (cid:88) j =1 e − κj (2 + 2 κj + κ j ) /j. (11)If only contribution from the two nearest neighbor par-ticles is retained ( j = 1), the conventional dust latticewave (DLW) sound velocity scale is obtained, c = c exp( − κ )(2 + 2 κ + κ ) . (12)Of course, transverse mode does not exist in truly 1Dcase. B. 2D case
Combining expressions for the sound velocities and re-duced excess energy and denoting M = u ex / Γ we get c l = c (cid:20) κ ∂ M∂κ − κ ∂M∂κ + 5 M (cid:21) , (13) c t = c (cid:20) κ ∂ M∂κ + κ ∂M∂κ − M (cid:21) . (14)The Madelung constant for the triangular lattice can bewell represented by M = − . . κ − . κ + 0 . κ + πκ . (15)In Eq. (15) it is taken into account that κ = √ πa/λ andΓ = (1 / √ π )( Q /aT ). The explicit expressions for thesound velocities are then c l = c (cid:18) . κ − . − . κ + 0 . κ (cid:19) , (16) c t = c (cid:0) . − . κ + 0 . κ (cid:1) . (17)The longitudinal sound velocity diverges as κ − / on ap-proaching the one-component plasma (OCP) limit, whilethe transverse sound velocity remains finite. C. 3D case
The relations between the longitudinal and transversesound velocities and the Madelung constant in 3D caseare c l = c (cid:20) κ ∂ M∂κ − κ ∂M∂κ + 4 M (cid:21) , (18) c t = c (cid:20) κ ∂ M∂κ + 2 κ ∂M∂κ − M (cid:21) . (19)The excess energy can be very well represented by theion sphere model (ISM) resulting in M = κ (cid:48) ( κ (cid:48) + 1)( κ (cid:48) + 1) + ( κ (cid:48) − e κ (cid:48) (cid:18) π (cid:19) / , (20)where κ (cid:48) = a/λ = κ (4 π/ − / and the last factor in(20) arises from Γ = ( Q /aT )(4 π/ − / in the presentnotation. The explicit expressions for the longitudinaland transverse sound velocities become c l/t = 115 (cid:18) π (cid:19) / c F l/t ( κ (cid:48) ) , (21)where, after some algebra, we obtain F l ( x ) = x (cid:2) (4 + 3 x ) sinh( x ) − x cosh( x ) (cid:3) [ x cosh( x ) − sinh( x )] , and F t ( x ) = x (cid:2) (3 + x ) sinh( x ) − x cosh( x ) (cid:3) [ x cosh( x ) − sinh( x )] . It will be shown below that c l diverges as κ − when theOCP limit is approached, while c t remains finite. III. MAIN RESULTSA. General trends
The calculated sound velocities are plotted in Figs. 1and 2. Δ / λ c t / c , c t / c l FIG. 2. Reduced transverse sound velocities of stronglycoupled Yukawa systems versus the screenening parameter κ = ∆ /λ . Velocities are denoted by solid curves. Dashedcurve show the ratio of longitudinal-to-transverse sound ve-locities. The blue (upper) curves correspond to 2D case. Thered curves are for 3D case. Figure 1 shows the longitudinal velocities for 3D, 2D,and 1D cases. In the weakly screened regime with κ (cid:46) κ → κ (cid:38) κ . This is a general property of steep repul-sive interactions, not based on the particular shape ofYukawa potential, and we will discuss this in more detailin Sec. III C.The transverse sound velocities plotted in Fig. 2 arefinite in the Coulomb limit and slowly decrease with in-crease of κ . The transverse velocity is somewhat higherin 2D than in 3D. The ratios c t /c l start from zero at κ = 0 and approach (cid:39) . κ increases to 5. This isyet another illustration of the strong inequality c l (cid:29) c t from the side of soft interactions, which has importantimplications in a broad physical context. B. Weakly screened limit
In the limit of the Coulomb gas, the longitudinal dis-persion relations do not exhibit acoustic asymptotes as k →
0. The dispersion relation in the absence of correla-tions (random phase approximation) can be obtained bysimply substituting g ( r ) = 1 in Eq. (1). This yields theconventional plasmon dispersion ω = ω p = 4 πQ n/m in the 3D case. In the 2D case the frequency grows asthe square root of the wave vector, ω ∝ k . In the 1Dcase random phase approximation produces an integralwhich diverges logarithmically at small r . This indicates c l / c c l / c Δ / λ c l / c FIG. 3. Reduced longitudinal sound velocity versus thescreening parameter κ = ∆ /λ . The panels from top to bot-tom correspond to 1D, 2D, and 3D cases, respectively. Solidcurves denote the weakly screened asymptotes, symbols cor-respond to the full calculation. The dashed curve for the 3Dcase is the fit from Ref. 9. that the longitudinal sound velocities should diverge onapproaching the κ → κ (cid:28) c l = c √ − κ ; (22)In 2D case we get c l = c (cid:18) . √ κ − . √ κ − . κ / (cid:19) ,c t = c (cid:0) . − . κ + 0 . κ (cid:1) ; (23)And, finally, in 3D case the sound velocities are c l = c (cid:18) . κ − . κ − . κ (cid:19) ,c t = c (cid:0) . − . κ + 0 . κ (cid:1) . (24)Alternative fits for the sound velocities in the 3D weaklyscreening regime have been previously suggested inRef. 9.The weakly screened asymptotes for the longitudinalmode (solid curves) are compared with the full calcula-tion (symbols) in Fig. 3. As the Coulomb κ → Δ / λ c t / c FIG. 4. Reduced transverse sound velocity versus the screen-ing parameter κ = ∆ /λ . The top (blue) curve and symbolscorrespond to the 2D case, the lower (red) curves and symbolsare for the 3D cases. Solid curves denote the weakly screenedasymptotes, symbols correspond to the full calculation. Thedashed curve for the 3D case is the fit from Ref. 9. is approached, the longitudinal sound velocity scales as c l /c ∼ √− κ ( D = 1), 2 . / √ κ ( D = 2), and3 . /κ ( D = 3). The last two coefficients are not just fit-ting parameters. It is known that in the weakly screeningregime (and only in this regime) the longitudinal soundvelocity does not depend on the coupling strength andtends to the conventional dust acoustic wave (DAW) ve-locity. The details can be found in Refs. 10, 54, and 55,here we just reproduce the scalings. In the 3D case wehave c DAW = ω p λ = (cid:114) πQ nm λ = c (cid:114) πκ (cid:39) c . κ . (25)Similarly, in the 2D case we get c DAW = ω p √ λ = c (cid:114) πκ (cid:39) c . √ κ . (26)It is observed that the weakly screened asymptoteswork quite well even outside the range of applicability,i.e. even at κ (cid:38)
1. The dashed curve in the bottompanel of Fig. 3 corresponds to the fit proposed in Ref. 9.The agreement is excellent for κ (cid:46) κ (cid:46) κ andremain finite in the limit κ →
0. We have c t /c (cid:39) . D = 2) and 0 .
440 ( D = 3). How this compares with theknown results for the one-component plasma (OCP) sys-tems with Coulomb interactions in 2D and 3D? For theOCP systems the transverse sound velocities are directlyrelated to the thermal velocity and the reduced excessenergy. In the 2D case we have c t = − v u ex . Combining this with the strong coupling asymptote, u ex (cid:39) − . Q /aT ), we get c t /c (cid:39) . c t = − v u ex . Using the ISM estimation of the OCP excess energy, u ex (cid:39) − ( Q /aT ) we get c t /c (cid:39) . κ (cid:46) C. Sound velocities for steep repulsive potentials
For steep repulsive potentials we should have | f (cid:48) ( x ) /x | (cid:28) | f (cid:48)(cid:48) ( x ) | . Then the main contribution tothe sound velocities comes from the second derivative ofthe potential. This main contribution to the longitudinalsound velocity can be evaluated from c l = c B l C D (cid:90) ∞ dxx D +1 g ( x ) f (cid:48)(cid:48) ( x ) , (27)where as usually in this paper x = r/ ∆. Further, forsteep interactions the main contribution to the integralabove comes from the first shell of neighbors at x (cid:39) x f (cid:48)(cid:48) ( x ) by f (cid:48)(cid:48) (1) underthe integral. Such substitution is exact only for a long-range logarithmic potential, but should provide a goodestimate for quickly decaying potentials and an RDF g ( x )that has a strong peak near x (cid:39)
1. The remaining of theintegral can be related to the number of nearest neighbors N nn using C D (cid:90) ∞ x D +1 g ( x ) f (cid:48)(cid:48) ( x ) dx (cid:39)C D (cid:90) x min x D− g ( x ) f (cid:48)(cid:48) (1) dx (cid:39) f (cid:48)(cid:48) (1) N nn , (28)where x min > g ( x ) (in the considered situation thevalue of the integral is not sensitive to x min , because themain contribution comes from the immediate vicinity of x = 1). Taking into account that at strong coupling N nn (cid:39)
12 ( D = 3), 6 ( D = 2), and 2 ( D = 1), we get c l = (cid:15)m f (cid:48)(cid:48) (1) , (1D) c l = 1816 (cid:15)m f (cid:48)(cid:48) (1) , (2D) c l = 1210 (cid:15)m f (cid:48)(cid:48) (1) . (3D) (29)Thus the, longitudinal sound velocities are all propor-tional to (cid:112) ( (cid:15)/m ) f (cid:48)(cid:48) (1), multiplied by a coefficient of or-der unity. This coefficient has the following scaling withthe dimensionality: 3D:2D:1D (cid:39) √ . √ .
13 : 1. The difference in the coefficients is insignificant taking into ac-count simplifications involved. This explains, why all thecurves approach the common asymptote as κ increases inFig. 1. This common asymptote is just the DLW nearestneighbor result of Eq. (12).Note that within this approximation the ratio of thelongitudinal to transverse sound velocities is c t /c l =1 / √ (cid:39) .
58, independently of dimensionality. Thedashed curves in Fig. 2 should approach this asymptoteas κ increases further. Note, however, that the QCA ap-proach itself cannot be applied for arbitrary large κ . Itloses its applicability when approaching the hard sphereinteraction limit. In the Appendix we discuss how the consideration inthis Section can lead to a simple freezing indicator, whichwas previously applied to various classical 3D fluids and,particularly successfully, to the 3D Yukawa fluid.
IV. CONCLUSION
The effect of spatial dimensions on the amplitude ofsound velocities in strongly coupled Yukawa systems hasbeen investigated. A unified approach, based on infinitefrequency (instantaneous) elastic moduli of fluids andisotropic solids has been formulated. In this approach,the sound velocities are expressed in terms of the excessinternal energy, which is very well known quantity forYukawa systems. Physically motivated expressions, con-venient for practical application have been derived andanalyzed. Relations to dust-acoustic wave (DAW) anddust-lattice wave (DLW) velocities have been explored.The regimes of weak and strong screening have been ana-lyzed separately. It has been demonstrated that at weakscreening ( κ (cid:46)
3) the longitudinal sound velocities indifferent spatial dimensions are well separated and theiramplitude increases with dimensionality. For strongerscreening ( κ (cid:38) ACKNOWLEDGMENTS
I would like to thank Viktoria Yaroshenko for readingthe manuscript.
Appendix: Related freezing indicator
To the same level of accuracy as in Sec. III C we canestimate the Einstein frequency in 3D systems with steepinterparticle interactions asΩ = n m (cid:90) d r ∆ φ ( r ) g ( r ) (cid:39) (cid:15)N nn m ∆ f (cid:48)(cid:48) (1) ∝ φ (cid:48)(cid:48) (∆) m . (A.1)The celebrated Lindemann melting criterion states thatmelting occurs when the particle root-mean-square vibra-tional amplitude around the equilibrium position reachesa threshold value of about 0 . (cid:104) ξ (cid:105) (cid:39) Tm Ω (cid:39) L ∆ , (A.2)where L is the Lindemann parameter. CombiningEqs. (A.1) and (A.2) we immediately see that at the fluid-solid phase transition one may expect φ (cid:48)(cid:48) (∆)∆ T (cid:39) const . (A.3)This kind of criterion was first applied to Yukawa sys-tems, in which case it works very well for κ (cid:46) and LJ-type systems, where it isable to approximately predict the liquid boundary of theliquid-solid coexistence region (freezing transition). Forpotentials, exhibiting anomalous re-entrant melting be-havior, such as the exp-6 and Gaussian Core Model, theagreement with numerical data is merely qualitative andits application is limited to the low-density region. From the derivation, it is expected that the freezing in-dicator (A.3) is more appropriate for steep interactions.Why it works so well for soft weakly screened Yukawasystems (including OCP), remains to some extent mys-terious. Note, however, that an alternative derivation ofthe freezing indicator (A.3) for Yukawa systems, basedon the isomorph theory approach, has been recently dis-cussed.
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