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Nature of vibrational eigenmodes in topologically disordered solids S. I. Simdyankin,1,2 S. N. Taraskin,3 M. Elenius,2 S. R. Elliott,3 and M. Dzugutov2 1Département de physique, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (Québec) H3C 3J7, Canada 2Department of Numerical Analysis and Computer Science, Royal Institute of Technology, SE-100 44 Stockholm, Sweden 2 3Department of Chemistry, University of Cambridge, 0 Lensfield Road, Cambridge CB2 1EW, United Kingdom 0 (Dated: February 1, 2008) 2 n Weusealocalprojectionalanalysismethodtoinvestigatetheeffectoftopologicaldisorderonthe a vibrationaldynamicsinamodelglasssimulatedbymoleculardynamics. Evidenceispresentedthat J the vibrational eigenmodes in the glass are generically related to the corresponding eigenmodes of 1 itscrystallinecounterpartviadisorder-inducedlevel-repellingandhybridizationeffects. Itisargued 1 that the effect of topological disorder in the glass on the dynamical matrix can be simulated by introducingpositional disorder in a crystalline counterpart. ] n PACSnumbers: 63.50.+x,63.20.Dj,61.43.-j n - s i I. INTRODUCTION canbeexpectedthatvibrationaleigenmodesintopologi- d cally disorderedstructures have approximately the same . t nature as in crystals with lattice disorder7,18. Namely, a Vibrational properties of disordered materials is one m of the current lively topics of modern condensed-matter they can be regarded as strongly hybridized crystalline physics1. Features of the disordered vibrational spec- eigenmodes,shiftedinfrequencyduetodisorder-induced - d trum, such as the boson peak, the Ioffe-Regel crossover level-repelling effects. In this paper, we present argu- n ments, based on a numerical analysis of a representative and vibrational localization are being investigated (see o e.g.2,3,4,5,6,7). topologically disordered structure, that this conjecture c holds true, at least, for the structure considered. [ Disorder in condensed matter can essentially be clas- sified into two basic types8: (i) lattice disorder in crys- 3 tals, e.g. in substitutional alloys, and (ii) topological v disorder in, e.g., glasses. Crystalline structures with lat- 6 The existence of a crystalline counterpart allows a 3 tice disorder can be studied analytically and consider- comparison to be made between a glass and its refer- 3 able progress has been achieved in understanding their ence structure and conclusions to be drawn about the 8 vibrational behaviour2,7,9. Structures with topological effect of topological disorder on the spectrum of vibra- 0 disorder have mainly been investigated numerically but tional excitations. Moreover, it is possible to compare 1 in some aspects analytically3,9,10,11,12,13,14,15. Different positional, e.g. quenched thermally-induced, disorder in 0 possible ways of describing the origin and nature of vi- / a crystal with topological disorder in the corresponding t brationalmodes,particularlyinthelow-frequencyrange, a glass in terms of their respective influence on the vibra- m in such disordered systems are still under debate. tional properties. In a computer simulation, positional The vibrational properties of crystals with lattice dis- - disordercanbeintroducedintoacrystallinestructureby d order can be successfully treated mainly because of the heating it in the course of a molecular-dynamics run up n existence of a well-defined reference structure, viz. the to a certain temperature below the melting point. Then o same crystalline structure but without disorder. The the dynamical matrix of an instantaneous configuration c choice of a reference structure for a topologically dis- : correspondingtothis temperaturecanbe calculatedand v ordered material is less obvious. In some cases, such analyzed. Thus it is possible to mimic the main features i a choice can be based on the existence of similar local X of the vibrational dynamics of topologically disordered order in both the topologically disordered structure and systemsbythevibrationaldynamicsoftheirpositionally r a its crystalline prototype. For example, α-cristobalite is disordered crystalline counterparts. In the following, we found to be a good crystalline counterpart for vitreous demonstratethatthiscanbedone,atleastforsomerep- silica16,17. The existence of similar structural elements resentative structures. inbothcrystallineandglassysystemsisexpectedtolead tosimilarvibrationaldynamics. Similaritiesinthevibra- tional dynamics within these structural units can be re- vealedbycomparingvibrationaleigenmodesofthetopo- The rest of the paper is arranged as following. The logicallydisorderedstructurewiththoseofitscrystalline local projectional analysis method is developed and de- counterpart. Suchacomparisoncanelucidatethenature scribed in Sec. II. The model is described in Sec. III. ofthedisorderedvibrationaleigenmodesandtheirpossi- Results and conclusions are given in Sec. IV and V, re- ble generic connection to crystalline modes. Basically, it spectively. 2 II. LOCAL PROJECTIONAL ANALYSIS to compile a histogram for a discrete set of equidistant values of ω for an arbitrary set of ω ≡ ω , i.e. for cryst µ dis A comparison of the vibrational dynamics in a glass each ωdis, and its counterpart crystalline phase can be based on a 1 local projectional analysis. Of course, the equilibrium A(ωcryst,ωdis)= A(ωk′β′,ωµ) . (4) N(ω ) atomic arrangements in the glass and the crystal are es- cryst kX′β′ sentially different, andthus we cannotexpandthe eigenmodes in a disordered system in terms of the crystalline Here k′β′ assume values for which ωcryst ≤ ωk′β′ < eigenmodes,aswecandofordisorderedlattices7. Never- ωcryst +dω, where dω is the width of a histogram bin, theless,whatwecandoinsuchasituationis tocompare and N(ωcryst) is the number of crystalline states in this the local atomic motions in topologically similar con- spectral interval of width dω. stituent structural units in different frequency ranges in the glass and the crystal. The local projectional analysis uses the information III. STRUCTURAL MODEL about atomic displacements contained in the vibrational eigenmodes {e } with eigenfrequencies {ω }. Here the As a representative example of a topologically disor- ξ,i ξ index i enumerates the atoms and the index ξ labels dered model structure, we consider a single-component the eigenmodes. Let us introduce the normalized lo- glasswithpredominantlyicosahedralorder(theICglass) cal displacement vector {u(l)} for a structural element constructed by means of molecular-dynamics simulation ξ,j l: u(l) =e /[p (ω )]1/2, where j enumerates the atoms with the use of a pair-wise interatomic potential19 (all ξ,j ξ,j l ξ quantities used in this paper are expressed in Lennard- within the structural unit, and pl(ωξ) = j|eξ,j|2. In Jones reduced units20, see Refs. 21,22 for more detail). the case of a crystal, an eigenmode is idePntified by the The 3N ×3N-dynamical matrix (the number of parti- wavevector k and the dispersion branch β. For a crys- cles is N = 16000) for a glassy minimum-energy config- talline structural unit l, a disordered structural unit m, uration has been calculated and diagonalized, yielding a crystalline eigenmode kβ and a disordered eigenmode allvibrationaleigenvectors,{e },andeigenfrequencies, µ,i µ, we can define the squared scalar product: {ω } for this system. A good crystalline counterpart for µ the IC glass is the σ phase, a Frank-Kasper crystal22,23. Al,m(ωkβ,ωµ)=|u(klβ)u(µm)|2 , (1) Forthelocalprojectionalanalysisofthetypedescribed above, we have chosen two Z14 Frank-Kasper polyhe- where uk(lβ), uµ(l) have 3Nu components, Nu being the dra (point group D6h), interpenetrating along the six- number of atoms within the structural unit. This quan- fold symmetry axes. This structural unit (Z14 ), com- 2 tity depends on the mutual orientationof the crystalline prisedof22atoms,representsashortsegmentofthe-72◦ and disordered structural units. We are interested in disclinationline21,24. Withintheunitcelloftheσ phase, finding the crystalline eigenmode which most resembles there are 16 interior atoms (centers of Z14 polyhedra) a given disordered mode and, therefore, we choose the of eight partially overlapping Z14 units. In the 16000- 2 maximum value (among all possible mutual orientations atom sample of the simulated IC glass, 236 such units preserving the symmetry) of the squaredscalarproduct, (some of them partially overlapping)covering 20% of all atoms have been identified. These atoms, together with A˜l,m(ωkβ,ωµ)=max{Al,m(ωkβ,ωµ)} . (2) their nearest neighbors, represent 52% of all atoms. ThenextstepistoaverageA˜l,m(ωkβ,ωµ)overallpossible disorderedand crystalline structural units, and thus cal- IV. RESULTS culate the averagedsquared scalar product, A(ωkβ,ωµ), A. Level-repelling effect 1 A(ωkβ,ωµ)= pl(ωkβ)pm(ωµ)A˜l,m(ωkβ,ωµ) , (3) P Xl,m Fig. 1 shows the calculated A(ω ,ω ) for dis- cryst dis ordered eigenmodes from the low-, middle- and high- where the local normalization factors, p , take into l(m) frequency parts of the spectrum (a set of crystalline account the weight of the corresponding structural unit eigenmodes uniformly covering the first Brillouin zone in the entire mode and P ≡ l,mpl(ωkβ)pm(ωµ). The with 103 points in k-space was used). As seen from this averagedscalarproductsA(ωkPβ,ωµ)canbe reducedto a figure, A(ω ,ω ), as a function of ω , represents cryst dis cryst functionoftwocontinuousvariablesA(ωcryst,ωdis)byan a broad distribution centered around ωcryst =ωcmryaxst, the interpolation. In this case, A(ωcryst,ωdis) quantifies the frequency of the crystalline spectrum for which the cys- degree of local similarity between one disordered mode talline modes have the greatest overlap with the modes withafrequencyaboutωdisandacrystallinemodewitha of the disordered system. This suggests that a disor- frequencyaboutωcryst. OnewaytoobtainA(ωcryst,ωdis) dered mode with a frequency about ωdis is generically from A(ωkβ,ωµ), which was used in the present work, is relatedtocrystallinemodesaboutωcmryaxst; inotherwords, 3 the first moment of this function: ω(1) (ω )= ωA(ω,ω )dω/ A(ω,ω )dω . (5) cryst dis Z dis Z dis _Α(ωω,)crystdis crystcrystg()ω IanndFibgy.2s(ubb)t,rωacc(1rty)isntg,caalbcauclaktgerdoubnodthfrboymthAe(dωicrreycstt,mωdeitsh)3o1d, is plotted versus ω . dis From Fig. 1, it is evident that the peak-shaped function, A(ω ,ω ), becomes increasingly broader with cryst dis 0 5 10 ω15 20 25 30 increasing frequency, ωcryst, meaning that there is an in- cryst creasingly weak correlation in character between modes inthedisorderedsystemandspecificsimilarmodesinthe FIG. 1: Averaged squared scalar product of the atomic dis- placementsofdisorderedandcrystallinelocalunitsrenormal- crystal. Suchalargebroadeningofthepeaksathighfre- ized to the maximum value as a function of the crystalline quencieshasalsobeenfoundinlatticemodels7subjectto frequency ω for three values of the disordered mode fre- large force-constant disorder and, in this sense, it is not cryst quency ωdis: •; 1.12, ◦; 6.84; ×, 16.81. Dashed line: vibra- surprisingand reflects the large degree ofdisorder in the tional density of states gcryst(ωcryst) for theσ phase. IC-glass. A similar effect is known for vibrational plane waves (analog of crystalline eigenmodes) propagating in glass. The distribution of the weights of different dis- ω(1) ordered eigenmodes contributing to a propagating plane cryst wave becomes increasingly broad with increasing plane- 0 5 10 15 20 20 20 wave frequency (the strong-scattering regime)5,6. The (a) (b) 18 18 shape of this distribution can even approach the shape 16 16 of the entire frequency spectrum of a glass. However, at 14 14 lowfrequencies,andspecificallyintheboson-peakregion, 12 12 ωBP ≃2.5,thepeakwidthofA(ωcryst,ωdis)israthernar- ω10 10ωdis row, indicating that there is quite a strong correlation between the two types of modes there (this is analogous 8 8 to the weak-scattering regime for plane waves5,6). 6 6 The deviation of the curve ω(1) (ω ) from the 4 4 cryst dis BP straight-line bisector, ω(1) = ω , reflects the level- 2 2 cryst dis repelling effect. It is clearly seen that the disordered 0 x 10−4 0 0.5 1 6 modes from the low-frequency region are pushed down x 10−3 (c) 2ω gdis(ω)/ω2 VH 24 ωg()/cryst iffnrreeqqfuureeeqnnuccyye,npcwayrhtialoesfttchhoeemsdppiaesrocetrdrduemrteodartmehesohdirefetsleadftreoudmpwctarhyrdestsahwlilgiinthhe- 0 0 5 1ω0 15 20 respect to ω(1) . cryst Why do we describe this phenomenon as a level- FIG.2: (a)Reducedvibrationaldensityofstates,gdis(ω)/ω2, repelling effect? One can argue alternatively that disor- for the IC glass showing the boson peak (BP). (b) Disorder- dered sytems are “softer” than their crystalline counter- induced level-repelling effect: the disordered mode frequency parts, and this results in a smaller sound velocity and in ωdisversusthecharacteristiccrystallinefrequencyωc(1ry)st eval- excess low-frequency eigenmodes which have nothing to uatedusingtwomethods: ◦,direct;•,background-subtracted do with the level-repelling effect. In the high-frequency with ω0 ≃ 20; ⊙, background-subtracted with ω0 ≃ 10, see range,theextramodescouldbeduetovibrationsinlocal Ref. 31. The arrows show the connection between the posi- structuralblockscharacterizedbyrelativelylarge(dueto tion of the lowest van Hove singularity (VH) and the boson statistical fluctuations) spring constants which are also peak (BP) via the local projectional analysis results. (c) Re- duced vibrational density of states for theσ phase. notrelatedtothelevel-repellingeffect. Thesearguments are not based on particular asumptions about the topol- ogy of the disordered system and therefore should be valid for disordered lattices as well. However, we have thevibrationalmotionoftheZ14 structuralunitsinthe demonstrated the relevance of the level-repelling effects 2 disorderedeigenmodeissimilartothatofthesamestruc- in describing vibrational spectra of disordered lattices tural units for the indicated eigenmodes in the σ phase. withforce-constantdisorder7. Thishasbeendonewithin Since A(ω ,ω ) as a function of ω is broad and the mean-field approximation (homogeneous disorder), cryst dis cryst asymmetric,particularlyathighfrequencies,weestimate ignoring the role of statistical fluctuations in the force- the characteristicfrequencyofthe crystallinemodes cor- constant distributions. We have explained the appear- responding to a disordered mode associated with ω as ance of the low-frequencyexcess modes (with the VDOS dis 4 x 10−3 3000 2.5 IC glass σ phase 2000 2 Debye for IC glass Debye for σ phase 3T 2ω1.5 C / v ω)/ 1000 g( 1 0.5 10−2 10−1 T 0 0 ↑ 5 10 15 20 25 30 ω FIG.4: Calculated valuesofthetemperaturedependenceof BP the vibrational contribution to the heat capacity for the IC FIG.3: Thereducedvibrationaldensitiesofstatesg (ω)/ω2 glass(solidline)andtheσ-phasecrystallineapproximant(cir- and g (ω)/ω2, for the IC glass and the σ phased,isrespec- cles),plottedasCv/T3. ThecurvefortheICglasscalculated cryst by omitting all the states with ω <4 (i.e. in the boson-peak tively. The position of the boson peak (BP) in the VDOS of region) is also shown by the dot-dashed curve. The calcu- theIC glass is marked by thearrow. lated Debye values for C /T3 for the IC glass (dashed line) v and σ-phase(dotted line) are also given. ∝ ω2) in terms of the well-known effect25 of the shift of the low-frequencypartofthe spectrumofquasi-particles correlationsinthevibrationalspectrumarenotpresently to lower frequencies due to level-repelling effects. The known. Another distinctive feature of glassy systems is mean-fieldapproachisveryaccurateinthelow-frequency the presence of topological disorder, the role of which in regime. This demonstrates the relative unimportance vibrationalspectra is not yet established. In our current of local soft regions (soft configurations) characterized analysis, we try to find out to what extent the ”lattice by relatively small force constants which are due to sta- ideology”(including the level-repelling effect) is applica- tistical fluctuations. These regions, of course, exist but ble to glassy models. the low-frequency eigenmodes are not localized at them duetothedelocalizedpropertiesofthevibrationalstates spectrum around zero frequency18,26. We have also ex- B. The boson peak aminedandconfirmedthisbymeansofmultifractalanal- ysis (to be published elsewhere). This is in contrast to Fig. 2(b) can be used to establish the nature of dis- the high-frequency part of the VDOS where the mean- ordered modes in different frequency ranges. The low- field approach describes well only the main features and frequencyregimeisofparticularinterestforvibrationsin tendencies in the spectrum, such as the shift to higher glasses, because of the so-called boson peak (BP) which frequencies (again due to level-repelling effects) but not occurs in the reduced VDOS, g (ω)/ω2, [see Figs. 2(a) dis thebandtailscontaininglocalizedstateswherestatistical and3] and in the temperature dependence of the (vibra- fluctuations of the force constants are very important. tional) heat capacity C , normalized by the Debye T3 v Thus,bearinginmindtheimportanceoflevel-repelling dependence [the solid line in Fig. 4]. effects in determining the spectrum of disordered lat- The vibrational contribution to the heat capacity per tices, we conjecture that a similar effect can also be im- particle is given by portant in topologically disordered structures. By this we mean level-repelling effects relative to the spectra 1 3N ~ωj 2 exp(~ωj/T) C (T)= . (6) of corresponding crystalline counterparts of disordered v N (cid:18) T (cid:19) [exp(~ω /T)−1]2 Xj=1 j sytems which are caused by positional and topological disorder. The crystalline counterparts are characterized Since all quantities in this paper are expressed in by the same local order as in the corresponding disor- Lennard-Jones reduced units, we use the value of ~ ≃ deredstructures. Intopologicallydisorderedsystems,the 0.02959 referred to argon. The Debye approximation of atoms in local structural units are displaced from their the above equation is given by ideal crystalline positions and the values of the force- constants are distributed, giving a situation resembling 12π4 T 3 CD(T)= , (7) latticemodelswithforce-constantdisorder. However,the v 5 (cid:18)θ (cid:19) D force constants of bonds emanating from a given central atomcanbecorrelatedwitheachother(seealsoRef.27). where θ = ~ω = 2π~(9N/4πV)1/3c is the Debye D D D Thissituationisincontrasttolatticemodelswithuncor- temperature28. The Debye velocity of sound c is de- D related force constants. The strength and role of these fined as c =(1/c3+2/c3)−1/3, where c and c are the D l t l t 5 longitudinalandtransversesoundvelocitiesrespectively, 1 which can be estimated from the slopes of the acoustic dispersion branches21. The Debye velocities for the IC 0.95 glass and the σ phase are cdis ≃ 3.43 and ccryst ≃ 4.74, D D respectively. )dis0.9 optic In Fig. 4, the peak in the curve of Cv/T3 versus T ω(pt iisndthuee vtoibrsatatitoensailnsptheectbruomso.n-Ipnedaekedr,anifgweeofcufrteqthueenlcoiwes- ), wso ~~ ~~ di frequencypartofthespectrum,includingthebosonpeak, ω (c0.1 then the peak in Cv/T3 is significantly suppressed (see wa acoustic the dash-dotted line in Fig. 4). The heat capacity of the 0.05 IC-glasscanbeeasilycomparedwiththatoftheσ-phase crystal. ItcanbeseenfromFig.4thattheplotofC /T3 v 0 for the σ-phase crystal[the circles in Fig. 4]tends to the 0 ↑ 5 10 15ω 20 25 30 35 Debye limit (the dotted line) at low temperatures. The BP dis increase above this level at higher T is due to the influ- FIG. 5: Hybridization parameters for acoustic (◦) and optic ence of van Hove singularities in the crystalline VDOS. (•) branchesversusthedisordered modefrequency ω . The dis This peak for the σ-phase crystal closely resembles the boson-peak (BP) frequency is marked byarrow. peakforthe IC-glass,whenthe modesinthe boson-peak region of the VDOS are omitted from the calculations. However, the peak in the total curve of C /T3 for the v IC glass is significantly shifted to lower frequencies, and also approaches a higher value of the Debye limit (the Knowing w (ω ), we can say which crystalline branch β µ dashed line in Fig. 4) because of a lower value of the mainlycontributestoaparticulardisorderedeigenmode. sound velocity. In particular, this is of interest for the low-frequency Following the arrows in Figs. 2(a)-(c), it appears that regime, where only the acoustic branches exist in the the vibrational states in the boson-peak region mainly crystal, but where we could also expect an admixture correspondtocrystallinestatesinthevicinityofthelow- of low-lying optic branches in a glass due to hybridiza- est van Hove singularity in the σ phase. Although this tioneffects. Theresultsofacalculationofthe hybridiza- method of analysis is unavoidably approximate, it does tion parameter for acoustic, w = 3 w , and optic, ac β=1 β seemto confirmthe physicalpicture for the originofthe w = 3Nu.c.w ,branchesareprePsentedinFig.5(with boson peak previously found in the f.c.c. force-constant opt β=4 β disorderedlattice7 and in models of disorderedsilicon29. Nu.c. bePing the number of atoms in the crystalline unit cell; N =30 for the σ phase). Here we have averaged Assuming such a scenario for the boson-peak origin, u.c. the hybridization parameter w over all acoustic and all we can compare its position, ω ≃ 2.5, with the posi- β BP optic branches in order to demonstrate the relative role tion of the lowest van Hove singularity, ω ≃ 4, and VH ofthesetypes(acousticoroptic)vibrationsindisordered thus conclude that ω −ω ∼ ω , which is an indi- VH BP VH modes. AsseenfromFig.5,evenforthelowest-frequency cationofstrongdisorder. Inthismodel,levelrepellingof disordered modes, the contribution of optic crystalline low-frequency optic modes to even lower frequencies sig- modes is dominant [w (ω ≃ 1) ≃ 0.8] which indi- nificantly contributes to the excess mode density in the opt dis catesverystronghybridizationeffectswiththedominant boson-peak region (see below). number ofoptic modes. Preciselybecause ofsuchstrong hybridization, it is not possible to distinguish between the contributions of acoustic branches of different polar- C. Branch-hybridization effects izations (transverse or longitudinal). The branches of different polarization are distinct for small wavevectors The level-repelling effect in glass is accompanied by only, but this part of the dispersion is suppressedby the another general effect, namely strong disorder-induced dominant contribution from all other wave vectors for hybridization of crystalline eigenmodes. In this effect, which the polarizationis not well defined. It is also seen many crystalline eigenmodes contribute to a particular in Fig. 5 that the contribution of the acoustic branches disordered eigenmode (cf. the broad distributions of is maximum at low frequencies and then monotonically A(ωcryst,ωdis) in Fig. 1). Disorder-induced hybridiza- decays with increasing frequency (whereas the opposite tion also involves hybridization between different crys- behavior is, obviously, found for optic modes), thus in- talline vibrational branches. A quantitative characteris- dicating an enhancement of the hybridization with optic tic of such hybridization can be defined via the branch branches. Such behaviour is qualitatively different from hybridization parameter, wβ(ωµ): the plateau behaviour found at low frequencies in the f.c.c. disordered lattice7, where only acoustic modes ex- wβ(ωµ)= A(ωkβ,ωµ)/ A(ωkβ,ωµ) . (8) ist, and is due to the existence of the extensive optic Xk Xkβ spectrum in the crystalline counterpart of the IC glass. 6 3 betweenthese atoms. Similarly the trace of the diagonal (a) block for atom i describes the effective force constant, 2.5 κ = D , for interactions of this atom with all ii α iα,iα other aPtoms. Two conclusions can be made from a com- 2 parison of these distributions for the IC glass and the σ gg phase. First, the distributions for the glass can be imag- diaρdia1.5 inedasbeinggenericallyobtainedfromthoseforthecrys- talbybroadeningthecorrespondingδ-functions. Second, 1 this broadeningis strong enoughso thatno tracesof the 0.5 individual δ-functions remain; this is an indication of a strong degree of disorder. 00 0.2 0.4 0.6 0.8 1 A natural question is, can we actually reproduce the 60 distributionsfortheglassfromthecrystallineones? The (b) simplestwaytotrytodothisistointroducethermal(po- 50 sitional)disorderintheσphasebyincreasingthetemper- ature, and analyzing the dynamical matrix correspond- 40 ing to the instantaneous non-equilibrium state in which offdiag30 all the atoms are displaced about their crystalline equi- ρ librium positions (positional disorder). We have indeed 20 foundthat,forsuchaforce-constantdisorderedcase,the distribution of the dynamical matrix elements (cf. the 10 dashedandsolidlinesinFig.6(a)and(b))andtheVDOS forthethermally(positionally)-distortedσphaseandthe 0 −0.02 0 0.02 κ 0.04 0.06 0.08 ICglass22aresimilar. Thisimpliesthatthemainfeatures of the vibrationalspectrum in a topologically disordered FIG. 6: Distributions of the dynamical matrix elements, system can be essentially reproduced by introducing po- scaled by the spectral band width corresponding to the IC sitional disorder in its crystalline counterpart, at least glass,fortheICglassatT =0(solidline),σphaseatT =0.4 for the metallic-type system studied here. The existence (dashedline),andσphaseatT =0(bars). (a)Distributionof of topological disorder, as distinguished from positional thetracesofthediagonal3×3blocks. Solidcirclescorrespond disorderaboutequilibriumcrystallinepositions,doesnot to matrix elements of Z14 units in the IC glass. (b) Distri- 2 seemtoplayamajorroleindeterminingthecharacterof bution of the traces (with opposite sign) of the off-diagonal the vibrationalmodes, atleastin this case. However,we 3×3 blocks. havenotdirectlyaddressedthesituationwherethereare differences in the topological connections of atoms with constant force constants. D. Distribution of the matrix elements in the Wehavealsofoundthatthe distributionofthe matrix dynamical matrix elements for atoms belonging to Z14 units involved in 2 the present analysis is similar to the total distribution Both level-repelling and hybridization effects have (see Fig. 6(a)). This confirms the representative charac- demonstratedthe stronginfluence ofdisorderonproper- ter of the Z142 units for which we have performed the ties of disordered and crystalline modes, thus indicating local projectional analysis. a strong degree of disorder in the IC glass. A possible way to quantify the degree of disorder is to compare the distribution of the dynamical-matrix elements in the IC V. CONCLUSIONS glasswith that inthe σ phase (see Fig. 6). The positive- definite dynamical matrix consists of D ×D blocks (D To conclude, we have presented evidence that the vi- standsforthedimensionality,D =3inthiscase). Theel- brational modes in a topologically disordered glass are ementsofthediagonalblocksaresubjecttosum-rulecor- generically related to the eigenmodes in the correspond- relations with the elements of the off-diagonal blocks28. ing crystalline counterpart via disorder-induced level- Therefore,therearefourdistinctdistributionsofthema- repelling and hybridization effects. In particular, the trix elements: diagonal and off-diagonal elements in di- extra states in the low-frequency regime (boson peak) agonal and off-diagonal blocks30. In Figs. 6(a)-(b), we appear to correspond to crystalline states in the vicin- have plotted the representative distributions, ρddiiaagg and ity of the lowest van Hove singularities in the crystalline ρoff , for the traces of the diagonal elements in diagonal VDOS. We have proposed a way of defining the degree diag and off-diagonal blocks, respectively. The trace of the of disorder (weak or strong) by comparing the distribu- off-diagonal block (taken with opposite sign) for atom i tionofthedynamicalmatrixelementsforatopologically interactingwith atomj canbe associatedwith the effec- disordered structure with its crystalline counterpart. It tive force constant, κ = − D , for interactions appears that the main features of the vibrational spec- ij α iα,jα P 7 trum of a topologically disordered (metallic) glass may from the following Swedish research funds: Natural Sci- be reproduced by the vibrational dynamics of its posi- ence Research Foundation (NFR), Technical Research tionally (thermally) distorted crystalline counterpart. Foundation(TFR),andNetworkforAppliedMathemat- ics (NTM). S.N.T. is grateful to EPSRC for support. Acknowledgments S.I.S., M.E. and M.D. thank Trinity College, Cam- bridge, U.K. for hospitality, and acknowledge support 1 S. 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