Low-frequency vibrations of jammed packings in large spatial dimensions
Masanari Shimada, Hideyuki Mizuno, Ludovic Berthier, Atsushi Ikeda
aa r X i v : . [ c ond - m a t . s o f t ] O c t Low-frequency vibrations of jammed packings in large spatial dimensions
Masanari Shimada, ∗ Hideyuki Mizuno, Ludovic Berthier, and Atsushi Ikeda
1, 3 Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan Laboratoire Charles Coulomb (L2C), Universit´e de Montpellier, CNRS, 34095 Montpellier, France Research Center for Complex Systems Biology, Universal Biology Institute,University of Tokyo, Komaba, Tokyo 153-8902, Japan (Dated: October 17, 2019)Amorphous packings prepared in the vicinity of the jamming transition play a central role intheoretical studies of the vibrational spectrum of glasses. Two mean-field theories predict that thevibrational density of states g ( ω ) obeys a characteristic power law, g ( ω ) ∼ ω , called the non-Debyescaling in the low-frequency region. Numerical studies have however reported that this scaling breaksdown at low frequencies, due to finite dimensional effects. In this study, we prepare amorphouspackings of up to 128000 particles in spatial dimensions from d = 3 to d = 9 to characterise therange of validity of the non-Debye scaling. Our numerical results suggest that the non-Debye scalingis obeyed down to a frequency that gradually decreases as d increases, and possibly vanishes forlarge d , in agreement with mean-field predictions. We also show that the prestress is an efficientcontrol parameter to quantitatively compare packings across different spatial dimensions. I. INTRODUCTION
Amorphous solids represent a ubiquitous state of mat-ter. Despite their importance, understanding their prop-erties has been a challenge in condensed matter physicsfor a long time. However, recent studies on the jammingtransition have opened the door to fundamental progressto understanding the physics of glasses [1–10]. When apacking of athermal particles interacting through a repul-sive, finite-range potential is compressed, particles startto overlap with each other at a certain density, where thepacking acquires a finite pressure p and shear modulus G ,i.e., it becomes a solid. This is the jamming transition.Jammed systems can be considered as a simple modelsystem for glasses, and such models have enabled the con-struction of sophisticated theories [4, 5, 8–10] that rely onthe specific critical properties of the jamming transition.Close to the transition, mechanical and geometrical ob-servables display power-law dependences on the distanceto the jamming transition point [1, 2]. Usually, the pres-sure p or the excess packing fraction ∆ φ = φ − φ J areused to measure the distance from the jamming transi-tion, where φ J is the packing fraction at the transitionpoint. Both quantities are easy to control in simula-tions and experiments, and they obey the simple rela-tion: p ∼ ∆ φ . The shear modulus G follows insteadthe scaling law G ∼ p / close to jamming [1, 2], andthus, the system gradually acquires rigidity as pressureincreases above jamming. The contact number per par-ticle, z , characterizes the geometrical properties of suchpackings, and it becomes exactly twice the spatial di-mensionality d at the transition point, which can be un-derstood from the Maxwell criterion [11]. Defining theexcess contact number as δz = z − d , the scaling rela-tion δz ∼ p / then holds [1, 2]. In addition to the above ∗ [email protected] scaling relations, many other quantities show power-lawbehaviors, such as the radial distribution function andthe force distribution [1, 2, 12–14], for which accuratetheoretical descriptions are now available [15, 16].In addition, the vibrational properties of jammed sys-tems have attracted intense attention. One motiva-tion is that they are expected to shed new light onthe low-frequency vibrational properties of structuralglasses, which govern their low-temperature thermalproperties [17–19], the structural relaxation of super-cooled liquids [20, 21], and their mechanical failure un-der load [22–24]. In particular, the vibrational density ofstates (vDOS) g ( ω ), where ω is the frequency, is a centralquantity for characterizing the vibrational properties ofa material. Near the jamming transition point, a char-acteristic plateau g ( ω ) ∼ ω is observed [1–3, 6]. Theonset frequency of the plateau is denoted by ω ∗ , and thisonset frequency goes to zero as the system approachesthe jamming transition, with a power-law dependence of ω ∗ ∼ p / [3, 6]. Below ω ∗ , the vDOS shows a quadraticfrequency dependence g ( ω ) ∼ ( ω/ω ∗ ) [9, 25, 26]. Sincethis dependence is independent of the number of spa-tial dimensions d , it is different from the Debye law g Debye ( ω ) ∼ ω d − [17], except in the important casewhere d = 3. Hence, this is called the non-Debye scal-ing [25]. The non-Debye scaling does not seem to extendto zero frequency. Instead, below a certain frequency,spatially localized vibrations called quasilocalized vibra-tions (QLVs) coexist with plane waves, or phonons [26–28]. These QLVs obey a quartic power law [26–32],namely, g QLV ( ω ) ∼ ( ω/ω ∗ ) [26, 31]. As a result, partic-ularly in three-dimensional space ( d = 3), the non-Debyescaling manifests itself as a peak in the reduced vDOS g ( ω ) /ω , called the boson peak [33–35]. We remark thatsince the Debye law is g Debye ( ω ) ∼ ω for d = 3, thereduced vDOS is g ( ω ) /ω ∼ g ( ω ) /g Debye ( ω ).To explain these observations, two kinds of mean fieldtheories have been developed: the replica theory for aperceptron [10, 36, 37] and the effective medium theory(EMT) [8, 9, 38]. The former [10, 36, 37] addresses aperceptron model, which is considered to belong to thesame universality class as jammed materials. The lat-ter [8, 9, 38] maps a jammed solid onto a disordered lat-tice and then considers the resulting equations of motion.Both theories predict that the vDOS should become flatfor ω > ω ∗ and should exhibit the non-Debye scaling g ( ω ) ∼ ω for ω < ω ∗ . Both these theories obtain thenon-Debye scaling as a consequence of the marginal sta-bility of the system [9, 10, 39]. Mathematically, marginalstability translates into full replica symmetry breakingin the replica theory and an elastic instability in the ef-fective medium theory. In particular, when the systemis on the verge of instability, the non-Debye scaling be-comes gapless [9, 10], namely the scaling g ( ω ) ∼ ω ex-tends down to zero frequency and should dominate (andreplace) the usual Debye law for solids. These theoriestherefore predict that for three dimensions, the bosonpeak will not be a ‘peak’ in a marginally stable glass butthat instead, the reduced vDOS g ( ω ) /ω should take atlow ω a constant value which is larger than the Debyeprediction.However, simulations of three-dimensional systemshave found that the boson peak is, perhaps unsurpris-ingly given its name, just a peak. The non-Debye scalingdoes not extend to zero frequency, and instead, QLVs ap-pear at low frequency and dominate the low frequency be-haviour. This discrepancy between the simulations andtheories can be attributed to either of two incompatibil-ities between them. (1) The theories are of mean-fieldnature and are expected to work well only in the infinite-dimensional limit, whereas the simulations are performedin a finite number of dimensions (mostly, three). There-fore, the breakdown of the non-Debye scaling and the ap-pearance of the QLVs may be due to finite-dimensionaleffects. (2) Theories do not directly address the pack-ing of particles. The replica theory considers a percep-tron model, whereas effective medium theory focuses ona spring network on a disordered lattice. The real pack-ings of the particles may not be similar to these models,even in the infinite-dimensional limit.To understand the discrepancy between the simula-tions and theories, it is necessary to numerically accessthe full frequency range of the non-Debye scaling forlarge-dimensional systems, to see whether and how thevDOS converges to the theoretical prediction in the large d limit. Previously, Charbonneau et al. [25] studied vi-brations in packings of particles in d = 3–7 and providedevidence in favor of the existence of a region of quadraticnon-Debye scaling in these dimensions. Kapteijns etal. [32] studied d = 2–4 and established instead the exis-tence of the quartic law due to QLVs in these dimensions,using similar models and parameters Charbonneau et al. .Therefore, the validity range of the non-Debye scalingwas not accessed in these earlier studies, and the impor-tant question regarding the discrepancy with the theoryhas not been addressed.In this work, we study the vibrational properties of packings of up to N = 128000 particles in dimensions d = 3–9 and answer the questions raised above. Be-fore studying the dependence of the vDOS on the spatialdimensionality, we first consider the appropriate normal-ization of our control parameter. Although the excesspacking fraction ∆ φ and the pressure p are useful quan-tities in low-dimensional systems for characterizing thedistance from the jamming transition, their complicateddependence on d makes it difficult to compare differentdimensionalities. Instead, we use the prestress, e [5, 9],which we define shortly. This quantity is more easilynormalized and handled in different spatial dimensions.By analysing the excess contact number and the onsetfrequency for the characteristic plateau in the vDOS, weshow that this choice enables us to best compare pack-ings in different dimensions. We then extract the onsetfrequency of the non-Debye scaling and study its depen-dence on the spatial dimensionality. We find that the on-set frequency decreases with increasing d , suggesting thatthe non-Debye scaling region extends to lower frequencywith an increasing number of dimensions, at least up to d = 9. Our numerical results suggest that the vDOSconverges to the predicted form and that the non-Debyescaling becomes gapless even in real particle systems inthe infinite-dimensional limit.In Sec. II we present the model and methods used inthe present study. In Sec. III we present the numericalresults, and we discuss them in Sec. IV. II. MODEL AND METHODS
We generated monodisperse packings of particles ofmass m in a periodic cubic box. The particles interactvia a finite-range harmonic potential: φ ( r ) = ǫ (cid:16) − rσ (cid:17) H ( σ − r ) , (1)where r is the distance between two particles; ǫ and σ arethe characteristic energy and length scales, respectively;and H ( x ) is the Heaviside step function, i.e., H ( x ) = 1for x ≥ H ( x ) = 0 for x <
0. In this paper, we re-port the length, mass, and time in units of σ , m , and p mσ /ǫ , respectively. The considered spatial dimen-sionalities are d = 3, 4, 5, 6, 7, 8, and 9. For each d , weprepared packings of N = 8000, 16000, 32000, 64000 and128000 particles. The mechanically stable configurations(inherent structures) were generated via quenching fromrandom configurations using the FIRE algorithm [40].After obtaining the inherent structures and removingrattlers (with a contact number of less than d ), we carriedout a normal mode analysis. We calculated the dynam-ical matrix, which is the second derivative of the systempotential U = P i To discuss the vibrational properties of jammed sys-tems, we first have to specify a control parameter to ex-press the distance from the jamming transition. Usualchoices are the excess packing fraction, ∆ φ = φ − φ J or∆ φ/φ J , and the pressure, p [1–3, 6, 7, 25]. However, theexcess packing fraction is not convenient for our study be-cause φ J depends on d and on the preparation protocolsin a highly nontrivial manner [47, 48], making it diffi-cult to compare different dimensionalities and differentprotocols [49]. The pressure is also not a very convenientchoice because it is equal to the total virial divided by thevolume of the system [50], and the volume of a jammedsystem also depends on d in a very complicated way. Asa result, the pressure and excess packing fraction cannotbe well controlled for different dimensionalities.A natural choice for the control parameter is found byconsidering the dynamical matrix given in Eq. (2) [5, 9,38]. In Eq. (2), there are two terms in curly brackets.The first one is proportional to the second derivative ofthe potential φ ′′ ( r ) = 1, and the second one is propor-tional to the force − φ ′ ( r ) = 1 − r . The first term is alwayspositive and can be simply interpreted as the stiffness ofthe spring between i and j . The second term is nega-tive and destabilizes motions along the d − r ij . The competition between these twoterms is crucial for the stability of the system. Therefore,we define the ratio of the second term to the first term as e = ( d − h− φ ′ ( r ij ) /r ij φ ′′ ( r ij ) i ij = ( d − h /r ij − i ij ,where h•i ij is the average over all contacts. This ratiois usually called the prestress [5, 9, 38]. Note that theprestress is proportional to the pressure near the jam-ming transition with a fixed d . Since EMT predicts that ω ∗ ∼ δz ∼ e / [5, 9], the prestress is a more funda-mental quantity than the pressure is for discussing thescaling relation. Furthermore, the excess contact num-ber is of order d , and thus, a suitable normalization forit is δz/ d = z/ d − δz/ d and (b) the onset frequency of the plateau in the vDOS ω ∗ as functions of e . The former is measured for N =16000, and the latter is measured for N = 8000 [51].Since ω ∗ cannot be defined far from the jamming transi-tion, we do not have data for e & . 07 in Fig. 1(b). Inboth Figs. 1(a) and (b), the data collapse to a single mas-ter curve. For e . . 01, we obtain δz/ d ≈ . e / and ω ∗ ≈ . e / (solid lines), which work almost perfectlyin all dimensions. For e & . δz/ d and ω ∗ start todeviate from these power laws. However, the data stillcollapse to a single master curve even in this region, espe-cially in large d . This suggests that our normalization isvalid even far from the jamming transition point, wherethese quantities no longer follow a power-law scaling. Ateven larger e , the excess contact number exhibits a kink: − − − − . e / (b)10 − − − − − . e / (a) ω ∗ e δ z / d ed = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9Figure 1. (a) The excess contact number δz divided by 2 d as a function of the prestress e . (b) The onset frequency ω ∗ as afunction of the prestress. In both plots, the solid lines are proportional to e / , with the indicated numerical prefactors. the most visible case is at e ∼ . d = 3. This kink cor-responds to the crossover to “deeply jammed” solids, inwhich particles interact with their second nearest neigh-bors [52, 53].From now on, we use e as the control parameter todiscuss the vibrational properties of jammed systems invarious spatial dimensions. B. Vibrational properties Having identified the appropriate control parameter e ,we then generate packings with identical prestress values, e = 0 . 25, 0 . 2, 0 . 15, 0 . 1, 0 . 05, and 0 . 01 for dimensions3 ≤ d ≤ ≤ N ≤ N = 128000, we could prepareonly packings of e ≥ . 15 in d = 6, e = 0 . 25 in d = 7, e = 0 . 25 in d = 8, and e = 0 . d = 9.First, we plot the participation ratio P k in Fig. 2.Since the results are qualitatively the same for all e , weonly show the data for e = 0 . P k values, i.e., theyare extended. On the other hand, in the low-frequencyregime, the participation ratio gradually decreases, sig-naling the existence of QLVs. As the dimensionality in-creases, the vibrations become more extended, and theonset frequency where the vibrations start to be local-ized decreases, as previously observed in Ref. [25]. Thisimplies that the non-Debye scaling g ( ω ) ∼ ω , whichconsists of extended vibrations, may be obeyed over a broader range of frequencies towards small frequenciesas the number of spatial dimensions increases.To quantitatively investigate the non-Debye scaling,we calculated the vDOS and the CD. In Fig. 3, we plotthe reduced vDOS for all e in d = 6. In all cases, weobserve the plateaus ˜ g ( ω ) ∼ ω , which correspond to thenon-Debye scaling g ( ω ) ∼ ω . Interestingly, although thenon-Debye scaling was initially discussed in the contextof the jamming transition, it can be observed even for e & . 01, where the power-law relation between the ex-cess contact number and the prestress no longer holds, − − − − P k ω k d = 3 d = 5 d = 7 d = 9 Figure 2. The participation ratio as a function of the fre-quency for e = 0 . 1. Each point indicates an eigenmode. Weshow the data only in odd numbers of dimensions for visual-ization purposes. − − − − ˜ g ( ω ) ωe = 0 . e = 0 . e = 0 . e = 0 . e = 0 . e = 0 . Figure 3. The reduced vDOS in d = 6 for e = 0 . 25, 0 . 2, 0 . . 1, 0 . 05, and 0 . as shown in Fig. 1. This suggests the possibility thatthe non-Debye scaling of the vDOS is a robust feature ofamorphous solids, irrespective of the jamming transition.We will further discuss this point in Sec. IV. From thedata for e ≥ . 1, we can appreciate the full frequencydependence of the non-Debye contribution to the den-sity of states and estimate where it begins and where itends. Thus, in the following, we focus on the case e ≥ . e = 0 . 1. The data for the other valuesof the prestresses e are available in Appendix A.Figures 4(a) and (b) show the vDOS and the CD, re-spectively. These plots indicate that as d increases ata fixed e , the vDOS and the CD overall converge todimension-independent functions. This finding is con-sistent with the results for the scaling behaviors of δz/ d and ω ∗ in Fig. 1. We note that the two peaks of the vDOSat ω ∼ . . d = 3 disappear and merge to forma single broad peak in a large number of dimensions.To examine the non-Debye scaling in the low-frequencyregime, we plot the reduced versions of these functionsfor the same prestress, e = 0 . 1, in Figs. 4(c) and (d). Wecan clearly see that these functions depend on the dimen-sionality in the lowest-frequency region. The results in d ≥ ω & . 1, and a fit to the data toa plateau corresponding to the non-Debye scaling is con-vincing, even on a logarithmic scale. This implies thatthe prefactor of the quadratic non-Debye scaling doesnot depend on d for d ≥ 4, and is solely controlled by thevalue of prestress e .On the other hand, when we focus on ω . . 1, wesee that the non-Debye scaling region extends to lower frequencies with increasing d from d = 4 to 9. Since thequartic frequency dependence of the QLVs g QLV ( ω ) ∼ ω has been reported in a previous study [32], we show thecorresponding dotted lines of slope 2 in Fig. 4(c) and ofslope 3 in Fig. 4(d). These fits suggest that the QLVssurvive up to dimension d = 9, but appear at lower fre-quencies for larger d .To quantitatively study this behavior, we measured thefrequency width of the non-Debye scaling region. We ex-tracted the two frequencies at which the reduced distri-bution is smaller than its maximum by 10%, which wedenote by ω max and ω min (with the convention ω min <ω max ). We use a superscript g or C to specify the func-tion from which each of these frequencies was extracted,i.e., four frequencies are considered for each e : ω g min , ω g max , ω C min , and ω C max . In Fig. 5, we plot (a) ω C min , (b) ω C max , and (c) ω C max /ω C max as functions of d for e ≥ . ω g min and ω g max , which exhibit qualitativelythe same behaviors as ω C min and ω C max , are shown in Ap-pendix A. The value of ω C min decreases with increasing d for d & 4, whereas ω C max increases at small d and thenquickly saturates for d & 5. These results are consis-tent with the observations in Figs. 4(b) and (d). Fromthese two results, we conclude that the non-Debye scal-ing region applies over a broader frequency range withincreasing d . By dividing ω C min by ω C max , we clarify thistendency in Fig. 5(c). This plot shows that ω C max /ω C min increases for all e as the number of spatial dimensionsincreases. Therefore, we conclude that the non-Debyescaling region becomes broader for larger dimensionality.Although our data are limited to d ≤ 9, the non-Debye scaling region continuously extends with increas-ing dimensionality without any sign of saturation; thus,we expect that the vDOS of a jammed particle systemapproaches the gapless non-Debye scaling in the large-dimensional limit; namely, it converges to the form pre-dicted by effective medium theory [9] and by replica the-ory for a perceptron [10]. We are not aware of any theo-retical prediction for how fast the large d limit should bereached by increasing d , but the data presented in thiswork suggest that the convergence, even if real, is rathermodest as the frequency width of the non-Debye scalingseems to grow linearly with d . Similar convergences to-wards the large d limit in the context of mean-field theoryis not infrequent [55, 56]. IV. SUMMARY AND DISCUSSION In this work, we have numerically studied the low-frequency vibrational properties of jammed particles in d = 3–9 spatial dimensions. We first showed that theprestress e = ( d − h /r ij − i ij is an appropriate con-trol parameter for studying jamming scaling behaviorsin different dimensions. In particular, the excess contactnumber divided by 2 d , δz/ d , and the onset frequencyof the flat region of the vDOS, ω ∗ , in various dimensionswere shown to follow universal functions of the prestress . . . . . . . . . . . . . . 81 0 0 . . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 ˜ C ( ω ) ω Figure 4. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 1. For the data for theother prestresses e , see Appendix A. The solid lines in (c) and (d) are the frequency dependence of the non-Debye scaling. Thedotted lines in (c) and (d) have slope 2 and 3, respectively, indicating the frequency dependence of the QLVs, g ( ω ) ∝ ω . e ; near the jamming transition, δz/ d ≈ . e / and ω ∗ ≈ . e / work almost perfectly in any number ofdimensions. Then, by comparing the vDOS in differentdimensions at the same prestress e , we studied the di-mensional dependence of the vDOS in the low-frequencyregion. Our system sizes of N = 8000–128000 enabled usto capture the full frequency range of the non-Debye scal-ing g ( ω ) ∼ ω in d = 3–9. We found that the non-Debyescaling appears below ω ∗ in all dimensions and that thefrequency width of the non-Debye scaling region growswith increasing dimensionality without any sign of satu-ration. From these findings, we expect that the vDOS ofa real packing of particles converges to the gapless non-Debye scaling in the large-dimensional limit, thus fullysupporting the prediction of effective medium theory [9]and replica theory for a perceptron [10]. Related to this finding, two comments are in order.The first concerns the precise form of the dimensionaldependence of ω C max /ω C min . A packing of particles in fi-nite dimensions will include rattler particles that do notcontribute to the rigidity [2]. The presence of rattlersis a kind of finite-dimensional effect, and previous stud-ies have established that the fraction of rattlers decreaseswith increasing dimensionality [12]. This decrease is veryrapid, with the fraction following ∝ e − αd with a constant α [12]. Based on this observation, one might expect thatthe finite-dimensional effect in the vDOS should also van-ish exponentially with increasing d , i.e., that ω C max /ω C min should increase exponentially. However, we found thatthe dimensional dependence of ω C max /ω C min is not verydramatic, at least in d ≤ 9, and that the data are stillcompatible with a linear dependence on d . It would be . . . . . . . 81 (b)234 3 4 5 6 7 8 9(c) ω C m i n ω C m a x ω C m a x / ω C m i n d e = 0 . e = 0 . e = 0 . e = 0 . Figure 5. (a) ω C min , (b) ω C max , and (c) ω C max /ω C min as functionsof d . The data are connected by lines (this is not a fit). Forthe data for ω g min and ω g max , see Appendix A. interesting to determine whether ω C max /ω C min ultimatelygrows exponentially at d ≥ 10, although the computa-tional cost of such a study is beyond our reach for themoment.Second, our study established that the non-Debye scal-ing holds even far from the jamming transition point, asshown in Fig. 3 and discussed in the corresponding para-graph. This result suggests that the non-Debye scalingmay be more universal than discussed so far in the con-text of the jamming transition. In fact, not only thetheories for jammed solids [9, 10, 38] but also elastic-ity theory with a fluctuating elastic modulus [57, 58]predict a quadratic frequency dependence of the vDOSnear the BP frequency. The latter theory is not rootedin jammed materials and regards glasses as elastic con-tinua with a spatially fluctuating elastic modulus to de-scribe the universal behaviors of the low-frequency exci-tations [57, 58]. In this respect, it will be interesting tostudy whether amorphous solids with other potentials,such as the Lennard-Jones potential, also exhibit non-Debye scaling in large dimensions. This topic will beaddressed in future work. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI GrantNumbers 19J20036, 17K14369, 17H04853, 16H04034,18H05225, 19K14670, and 19H01812. This work wasalso partially supported by the Asahi Glass Foundation.The research leading to these results has received fundingfrom the Simons Foundation (Grant No. 454933, LudovicBerthier). Appendix A: Additional data for the vDOS and the CD We report additional data for the vDOS and the CD with different e values (supplementing Fig. 4) and for ω g min and ω g max (supplementing Fig. 5). . . . . . . . . . . . . . . 81 0 0 . . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 ˜ C ( ω ) ω Figure 6. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 25. Due to numericallimitations, we could not calculate the results for d = 9. The solid lines in (c) and (d) are the frequency dependence of thenon-Debye scaling. The dotted lines in (c) and (d) have slope 2 and 3, respectively, indicating the frequency dependence of theQLVs. . . . . . . . . . . . . . . 81 0 0 . . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 ˜ C ( ω ) ω Figure 7. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 2. The solid lines in (c) and(d) are the frequency dependence of the non-Debye scaling. The dotted lines in (c) and (d) have slope 2 and 3, respectively,indicating the frequency dependence of the QLVs. . . . . . . . . . . . . . . 81 0 0 . . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 ˜ C ( ω ) ω Figure 8. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 15. The solid lines in (c) and(d) are the frequency dependence of the non-Debye scaling. The dotted lines in (c) and (d) have slope 2 and 3, respectively,indicating the frequency dependence of the QLVs. . . . . . . . . . . . . . . 81 0 0 . . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 ˜ C ( ω ) ω Figure 9. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 05. The solid lines in (c) and(d) are the frequency dependence of the non-Debye scaling. The dotted lines in (c) and (d) have slope 2 and 3, respectively,indicating the frequency dependence of the QLVs. . . . . . . . . . . . . 81 0 0 . . . − − − (c) g ( ω ) ∝ ω g ( ω ) ∝ ω − − − − (d) C ( ω ) ∝ ω C ( ω ) ∝ ω g ( ω ) ω C ( ω ) ω ˜ g ( ω ) ωd = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 ˜ C ( ω ) ω Figure 10. (a) The vDOS, (b) the CD, (c) the reduced vDOS, and (d) the reduced CD for e = 0 . 01. The solid lines in (c) and(d) are the frequency dependence of the non-Debye scaling. The dotted lines in (c) and (d) have slope 2 and 3, respectively,indicating the frequency dependence of the QLVs. . . . . . ω g m i n ω g m a x ω g m a x / ω g m i n d e = 0 . e = 0 . e = 0 . e = 0 . ω g min , (b) ω g max , and (c) ω g max /ω g min as functions of d . The lines are simple guides for the eye. [1] Corey S. 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