Coarsening Dynamics of an Antiferromagnetic XY model on the Kagome Lattice: Breakdown of the Critical Dynamic Scaling
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Coarsening Dynamics of an Antiferromagnetic XY model on the Kagome Lattice:Breakdown of the Critical Dynamic Scaling Sangwoong Park, Bongsoo Kim, and Sung Jong Lee Department of Physics, The University of Suwon, Kyonggi-do 445-743, Korea Department of Physics, Changwon National University, Changwon, 641-773, Korea
We find a breakdown of the critical dynamic scaling in the coarsening dynamics of an antifer-romagnetic XY model on the kagome lattice when the system is quenched from disordered statesinto the Kosterlitz-Thouless ( KT ) phases at low temperatures. There exist multiple growing lengthscales: the length scales of the average separation between fractional vortices are found to be not proportional to the length scales of the quasi-ordered domains. They are instead related through anontrivial power-law relation. The length scale of the quasi-ordered domains (as determined fromoptimal collapse of the correlation functions for the order parameter exp[3 iθ ( r )]) does not follow asimple power law growth but exhibits an anomalous growth with time-dependent effective growthexponent. The breakdown of the critical dynamic scaling is accompanied by unusual relaxationdynamics in the decay of fractional (3 θ ) vortices, where the decay of the vortex numbers is charac-terized by an exponential function of logarithmic powers in time. PACS numbers: 64.70.qj, 64.60.Ht, 75.10.Hk
November 8, 2018
I. INTRODUCTION
Thermodynamic systems quenched from a high-temperature disordered phase into a low-temperature or-dered phase exhibit characteristic growth of the lengthscale ℓ of the ordered domains which in typical situa-tions can be represented as a power law ℓ ∼ t /z , wherethe growth exponent 1 /z depends on the dimension ofthe space and of the relevant order parameter as wellas on the conserved or nonconserved nature of the orderparameter. Phase ordering dynamics (or coarsening process) isusually accompanied by the annihilation and decay ofthe characteristic topological defects such as point vor-tices or domain walls which are generated in the initialdisordered states. One of the most important notions inunderstanding and analyzing these coarsening processesis the so-called dynamic scaling hypothesis for the equal-time spatial correlation function of the order parameter,which is closely related to the observed self-similarity ofthe coarsening systems at different time instants.When the low temperature phase of the system is char-acterized by a quasi-long-range order with power law de-cay of the spatial correlation function of the order pa-rameter at equilibrium, then the dynamic scaling is gen-eralized to the critical dynamic scaling . One of the wellknown examples is the ferromagnetic XY model on thesquare lattice.However, it should be noted that this (critical) dy-namic scaling hypothesis for the phase ordering (or quasi-ordering) dynamics has not been proved on some generaltheoretical basis. Therefore, it is not clearly known whatis the condition for the validity of the dynamic scalingwhich is usually assumed in analyses of experimental re-sults or numerical simulations on coarsening dynamics. In typical situations, coarsening dynamics is investi-gated on systems where the phase ordering is accom-panied by the breaking of discrete or continuous globalsymmetry. However, there exist model systems that ex-hibit infinite ground state degeneracies of a discrete na-ture in addition to the usual (global) continuous sym-metry. Prominent examples are geometrically frustratedspin systems such as antiferromagnetic Ising models on atriangular lattice , antiferromagnetic XY or Heisenbergmodels on kagome or pyrochlore lattices. These sys-tems exhibit interesting equilibrium and nonequilibriumbehavior due to the geometric frustration effect, includ-ing spin-glass-like relaxation dynamics without quencheddisorder.In this work we perform numerical simulations on thecoarsening of an antiferromagnetic XY model on thekagome lattice ( KAFXY model) which is oneof the simplest geometrically frustrated models with in-finite ground state degeneracies. Experimentally, thismodel can be realized in superconducting Josephson-junction arrays or superconducting wire networks on a kagome lattice when a perpendicular magnetic fieldof half flux quantum (per plaquette) is applied on thesystem. We can also find physical examples in theanisotropic limit of Heisenberg antiferromagnets on thekagome lattice. It is well known that the system ex-hibits an infinite ground state degeneracy with finiteentropy . The system also exhibits a finite tempera-ture KT transition corresponding to the unbinding ofso-called fractional 3 θ -vortices. That is, at low tempera-ture below the KT transition, equilibrium of the systemwill be characterized by quasi-long-range order of the or-der parameter ψ ≡ e iθ ( r ) . Analogous to the case ofsimple ferromagnetic XY model on a square lattice, wemight expect that the coarsening dynamics of the systemwould exhibit a critical dynamic scaling for the equal-time spatial correlation of the order parameter.Our simulations, however, show that the critical dy-namic scaling is not obeyed very well at least for thetime duration of our numerical simulations ranging sev-eral decades of time scale. That is, it was not possible toachieve a good scaling collapse for the equal-time spatialcorrelation functions of the 3 θ order parameter. Anothersignature of the breakdown of the critical dynamic scal-ing is that the length scales corresponding to the aver-age separation between fractional vortices are not propor-tional to the length scales of the growing quasi-ordereddomains, which are instead related through a nontriv-ial power-law relation. This means that the two lengthscales exhibit different growth behavior in time. In termsof the decay of the fractional vortices, we also found thatthe fractional 3 θ -vortices residing in the small triangularplaquettes exhibit faster decay, while, in contrast, thosevortices sitting on the larger hexagonal plaquettes exhibitmuch slower decay that can be fitted by an exponentialof logarithmic powers in time .In addition to these features of multiple length scalesin the coarsening dynamics, the time dependence of thelength scale of the quasi-ordered domains does not ex-hibit a simple power law growth but rather exhibits ananomalous growth with time-dependent effective growthexponent. This appears to be closely related to the un-usual relaxation dynamics in the non-power-law decay ofthe fractional (3 θ ) vortices. It is not clear yet whetherthe scaling may be restored in the asymptotic limit. II. THE MODEL SYSTEM AND SIMULATIONMETHODS
The kagome lattice consists of corner-sharing trian-gles (Fig. 1). In an antiferromagnetic XY model on thekagome lattice, the Hamilitonian is defined as H = − J X h i,j i cos( θ i − θ j ) . (1)where J <
0, the sum runs over all nearest neighborpairs of sites, and θ i denotes the angle of the planar spinat site i with respect to some fixed direction in the twodimensional spin space. h i, j i indicates all pairs of nearestneighbor sites in the kagome lattice.It is easy to see that the ground states of this systemhave the property that for all pairs of nearest neighbors i and j , the angle difference satisfies | θ i − θ j | = 2 π/ KAFXY model are equivalent (up to a globalrotation) to the ground states of the three state Pottsmodel on the kagome lattice. In Figs. 1(a)-(b) are showntwo examples of simple ground states with long-range or-der, the so-called q = 0 state and q = √ × √ q refers to the wave vector corresponding to theperiodicity of the chirality configuration. It is easy to see,however, that in addition to these ground states with sim-ple spatial order there also exist infinitely many groundstates with no spatial order. It is well known that the system has a ground state entropy of S ≃ . k B persite . Now, if we consider the angle variable 3 θ and thecorresponding complex order parameter Ψ ≡ exp( i θ ),the degenerate ground states are all completely orderedin terms of this new order parameter. And it has beenshown that the system undergoes a KT transition at afinite temperature, where the spatial correlation of theorder parameter exhibits an algebraic decay below thetransition temperature.As for the KT transition temperature of this system,an analytic approximation was given as T c = π √ J ≈ . J. (2)Numerical simulations were performed by Rzchowski where he found two slightly different estimates on thetransition temperature, i.e., one estimate of T c ≃ . J based on the Binder cumulants of the order parameterand another of T c ≃ . J from the helicity modulus(or correspondingly the decay exponent η ( T c ) = 0 .
25 ofthe spatial correlation of the order parameter).In the present work, the coarsening dynamics of themodel system is performed via kinetic Monte Carlo meth-ods with standard Metropolis algorithm for quenches tovarious temperatures near and below the KT transition.System sizes ranging up to N × N = 256 ×
256 were em-ployed with periodic boundary conditions. In a kagomelattice, the number e N of the total spins is e N = N .The system is quenched from completely disordered ini-tial states down to a given low temperature with the pro-cess of coarsening being monitored through equal-timespatial correlation functions and the decay of topologicaldefects, etc.In addition to quenches from disordered state to lowtemperatures, we also performed the so-called nonequi-librium relaxation by suddenly bringing the system fromone of the ground states to some target temperaturesaround or below the KT transition. This method wasfound to be convenient for measuring the values of thecritical exponent η for the equilibrium spatial correlationfunctions.When we let the systems evolve from random initialconfigurations the following quantities can be measured:1. The number density n v ( t ) of topological defectswhich are 3 θ -vortices in the KAFXY model n v ( t ) ≡ h N v ( t ) i ˜ N , (3)where N v ( t ) is the total number of 3 θ -vortices (bothpositive and negative) at time t and ˜ N denotes thetotal number of sites (i.e., spins). h· · · i denotesan average over random initial configurations. Wealso count the separate number density of vorticesresiding on the hexagonal plaquettes ( n h ) and thoseon the triangular plaquettes ( n t ).2. The equal-time spatial correlation function of the3 θ order parameter ψ ( r , t ) ≡ e iθ ( r ,t ) . C ( r, t ) = h ψ ∗ ( r , t ) ψ (0 , t ) i (4)= 1 e N *X i exp(3 iθ i ( t ) − θ i + r ( t )) + (5)3. Nonequilibrium spin autocorrelation functions A ( t ) = 1 e N *X i exp( iθ i (0) − θ i ( t )) + (6) A ( t ) = 1 e N *X i exp(3 iθ i (0) − θ i ( t )) + . (7) III. SIMULATION RESULTS
We have performed dynamic Monte Carlo simulationsof
KAFXY model on a kagome lattice of dimensions256 × η for the spatial decay of the equi-librium spatial correlation function C eq ( r ) of the 3 θ or-der parameter, we have employed the so-called nonequi-librium relaxation (NER) method, where the system issuddenly brought from ground states to finite tempera-tures below or near the KT transition. Simulations wereperformed up to 655360 Monte Carlo steps which wassufficient for the equilibrium to be attained. This wasconfirmed by the collapse of the spatial correlations atlater time stages. In Fig. 2(a) the correlation functions C eq ( r ) are displayed on a log-log scale. The values ofthe exponent η thus determined for temperatures rang-ing from T = 0 .
01 to T = 0 .
074 are shown in Fig. 2(b)as well as in Table I.
T /J η ( T ) T /J η ( T )0.01 0.047(3) 0.065 0.254(11)0.02 0.082(4) 0.066 0.259(13)0.03 0.116(5) 0.067 0.264(12)0.04 0.153(8) 0.068 0.268(12)0.05 0.198(7) 0.069 0.280(11)0.06 0.235(9) 0.070 0.300(13)0.061 0.237(10) 0.071 0.308(12)0.062 0.240(11) 0.072 0.310(13)0.063 0.245(10) 0.073 0.330(15)0.064 0.250(12) 0.074 0.340(13)TABLE I: The equilibrium exponent η for different tempera-tures for the power-law decay of the spatial correlation of the3 θ order parameter. As for the generation of the initial low energy con-figurations, we first select one of the three phases of 0, ± π/ T = 0 . NER in order to bring the system to the equilibrium atsome given target temperature.From the graph of η ( T ) shown in Fig. 2(b), we findthat the exponent increases almost linearly in tempera-ture up to around T ≃ . η ( T ) ≃ . T . We also find that the expo-nent η takes the value of 1 / T ( η = 1 / ≃ . J (Table I). This temperature is expected to correspond tothe KT transition, which apparently is a little lower thanthe theoretical or the numerical estimates of earlierworks mentioned above.Now, the coarsening dynamics of the model systemunder quench to low temperature from disordered ini-tial states is investigated by monitoring the equal-timespatial correlation functions of the ψ order parameter.One of the most important features of typical coarsen-ing dynamics toward quasi-ordered phase is the criticaldynamic scaling C ( r, t ) = r − η ( T ) f ( r/L ( t )) , (8)where f ( x ) is the scaling function and L ( t ) is the growinglength scale. It should be noted that the above scalingansatz is based on the existence of a single growing lengthscale L ( t ).Figure 3(a)-(b) shows the equal-time correlation func-tions C ( r, t ) of KAFXY model for different time instants(from t = 10 to 655360) at the temperature T = 0 . η ( T ) (de-termined as in Table I) and evaluate the combination e C ( r, t ) ≡ r η ( T ) C ( r, t ). And then, for a given time in-stant, the length scale L ( t ) is determined by the condition e C ( r = L ( t ) , t ) = C where the constant C is chosen as C = 0 . C = 0 . e C ( r, t ) in terms of the rescaled dis-tance r/L ( t ), we can check whether the critical dynamicscaling holds or not from the quality of the collapse ofthe scaled correlation functions.By this procedure, we found, rather unexpectedly, thatthe critical dynamic scaling does not hold in the coarsen-ing dynamics of KAFXY model, at least for the (MonteCarlo) time duration of our simulations (up to 655360MC steps). The result of the scaling attempt is shown inFig. 3(c) for the case of T = 0 .
06 where we can see thatthe critical dyanmic scaling is not obeyed. We also trieddifferent values of η for the rescaled correlation with nosuccess. It might be possible that the scaling is restoredin the limit of infinitely long time.Even though the critical dynamic scaling is not faith-fully obeyed, we may still extract approximate lengthscale L ( t ) of spatial correlation in the manner describedabove. The length scale L ( t ) thus obtained exhibitsa rather unusual time-dependent behavior (Fig. 3(d)).That is, in the early-time stage up to around t ∼ , L ( t )exhibits a slow growth, which appears approximately in-dependent of the temperature. However, in the inter-mediate and late time stage of t & , the growth ofthe length scale L ( t ) becomes strongly dependent on thetemperature. In addition, for a given temperature, nosimple power law is found which is valid for the wholelate time regime. Instead the local logarithmic slope ex-hibits a steady increase in time in the late time regime.We can investigate the time-dependence of the growthof the length scale by defining the effective local growthexponent as β ( t ) ≡ d ln( L ( t ) /d ln( t ). Since we took thegrowing length scale L ( t ) only for discrete time instants t i with fixed interval in logarithmic scale, we evaluate thediscrete version of the above logaritmic slope as β ( t ′ i ) ≃ ln( L ( t i +1 ) /L ( t i ))ln( t i +1 /t i ) (9)where t i ≡ i , i = 1 , , · · · and t ′ i ≡ √ t i t i +1 . Fig-ures. 3(e)-(f) show the effective growth exponent at time t (for the case of C = 0 . C = 0 . T & . . ∼ . θ -vortices residing on the triangular plaquettes as well ason the hexagonal plaquettes of the kagome lattice. Theresults are shown in Figs. ?? (a)-(d) where we can seethat, for a given temperature, the triangular vortices aredecaying much faster than the hexagonal vortices. Wealso find that in general (for both types of the vortices)the decay of the total vortex numbers do not exhibit asimple power law behavior valid for the whole time range.One prominent feature in terms of the temperature de-pendence of the decay of the fractional vortex density isthat, for the vortices on the triangular plaquettes, thedecay rate increases as the temperature is lowered, whileon the other hand, those vortices on the hexagonal pla-quettes exhibit opposite dependence on the temperaturewith the decay rate decreasing sharply as the tempera-ture is lowered. This can be interpreted as implying that there is al-most no (free) energy barrier for the motion and decay ofthe triangular vortices but that some finite barrier existsfor the hexagonal vortices. We also find an interestingfeature of the hexagonal vortices (at lower temperatures)where, at initial stage, they increase a little and thenstart to decay. This can probably be understood as theinfluence of the decay of the neighboring triangular vor-tices with their excess energy turned over to neighboringhexagonal vortices, thus generating hexagonal vortices.Therefore, we can conclude that, for wide range of tem-peratures, the relaxation of the fractional vortices resid-ing on the hexagonal plaquettes determines the coars-ening process such as the correlation length scale cor-responding to the quasi-ordered domains. In addition,the vortex relaxations exhibit a considerable deviationfrom power law behavior with slowly increasing (in abso-lute magnitude) local logarithmic slope in the late timeregime. Especially in the case of the hexagonal vortices,the decay of the vortex number density could be fit to aform with N h = N exp[ − b (ln( t )) α ] (10)= N exp (cid:20) − (cid:18) ln( t )ln( t ) (cid:19) α (cid:21) . (11)with b ≡ ln( t ) − α . Here, we found that typically α takesvalues around 3 . ∼ . ). Wefound that it could also be fitted to an stretched expo-nential form as N h ( t ) ∼ N exp[ − ( t/t ) α ] , (12)with α ≃ . ± . ξ t ≡ / √ n t which corresponds to the average sep-aration between the vortices on the trianular plaquettes.Similarly, we can define the length scale ξ h ≡ / √ n h corresponding to the average separation between vorticeson the hexagonal plaquettes. We also define the lengthscales ξ v ≡ / √ n v = 1 / √ n h + n t which represents thelength scale corresponding to the total number densityof vortices.Figure 5 shows the three length scales at T = 0 . ?? (a)-(d) comparison is madebetween the length scales L ( t ) derived from the spatialcorrelation functions and the length scales ξ v derivedfrom the total vortex densities. We see that (for bothcases of C = 0 . C = 0 .
3) there exists no simpleproportional relationship between the two length scales.Rather, the length scale L ( t ) from the spatial correlationis seen to grow faster than the vortex lenth scale ξ v . Thisprobably implies that the vortices are not distributedevenly (statistically speaking) in the system but that thevortices are somehow distributed in a non-random man-ner such that some degree of clustering occurs. Interest-ing result is that the two length scales satisfy some non-trivial relationships such that ξ v ∼ L ( t ) λ with λ ≃ . C = 0 . ?? (b)) and λ ≃ .
68 for thecase of C = 0 . ?? (d)). This means that the vor-tex configuration of the system do not exhibit simple self-similarity at different time instants. Rather the vorticesprobably tend to cluster more unevenly as time passes by,leading to the correlation length scale L ( t ) growing fasterthan the length scale derived from the vortex density.The snapshot of defect configurations for the case of T = 0 .
06 is shown in Fig. ?? (a)-(d). We can easily rec-ognize faster decay of those vortices on the triangularplaquettes. From the snapshots alone, however, it is noteasy to detect the tendency of relative clustering of thevortices in the late time regime.Now we turn to the autocorrelation functions of thevariables exp( iθ ) and exp(3 iθ ). For these two variables,we find that the simulation results on the autocorre-lations are very different. For the case of exp( i θ ),we obtain a approximately power-law decay behavior A ( t ) ∼ t − λ in the early time regime up to around t ≃ . The value of the exponent λ ranges from λ ≃ .
56 (for T = 0 .
01) to λ ≃ .
90 (for T = 0 . A ( t ), we could not at-tempt a comparison of A ( t ) with the growing lengthscale L ( t ).In contrast, in case of the phase exp( iθ ), the spin au-tocorrelaton exhibits less statistical fluctuations with anon-power law behavior that can be reasonably fitted by A ( t ) = A exp[ − b (ln( t )) γ ] (13)= A exp (cid:20) − (cid:18) ln( t )ln( t ) (cid:19) γ (cid:21) (14)with the exponent γ = 2 . ± . b ≡ ln( t ) − γ . Oneof the simulation results is shown in Fig. ?? (a-b) for thecase of T = 0 .
06 with e N , which shows a suitable fit tothe above functional form with b ≃ .
15 and γ ≃ . γ = 1, A ( t ) reduces to a powerlaw of A ( t ) ∼ A t − b . The fitted values of γ implies that A ( t ) exhibits a considerable deviation from a power-lawbehavior. We also attempted to fit the simulation resultsto stretched exponential form. This resulted in rathersmall values for the stretching exponent of α ≃ .
14. Itis interesting to note that similar behavior of relaxationin the autocorrelation function of the order parameterwas reported in the coarsening dynamics of the so-calledHamiltonian XY model on a square lattice. In summary, we investigated the coarsening dynam-ics of the antiferromagnetic XY model on a kagome lat-tice. We found that the critical dynamic scaling is vio-lated. Novel and unusual coarsening dynamics may beattributable to the existence of infinite ground state de-generacies leading to nontrivial decay behavior of thedensity of defects residing on the triangular and hexago-nal plaquettes with nontrivial growth of multiple lengthscales. The competition between the critical thermal fluc-tuations of the equilibrium and the infinite ground statedegeneracy probably gives rise to the nontrivial featuresof the relaxation dynamics. It may be possible to collapsethe equal-time spatial correlation functions through amultiscaling scheme which we haven’t tried yet. It wouldbe also interesting to investigate the aging dynamics ofthis model system. A. J. Bray, Adv. Phys. , 357 (1994). V. L. Berezinskii, Sov. Phys. JETP , 610 (1972). J. M. Kosterlitz and D. Thouless, J. Phys. C , 1181(1973); J. M. Kosterlitz, J. Phys. C , 1046 (1974). E. Kim, B. Kim, and S. J. Lee, Phys. Rev. E , 066127(2003) and references therein. E. Kim, S. J. Lee, and B. Kim, Phys. Rev. E , 021106(2007). I. Syozi, Prog. Theor. Phys. , 306 (1951). R. Moessner and A. P. Ramirez, PHYSICS TODAY,February 2006, page 24. D. A. Huse and A. D. Rutenberg, Phys. Rev. B , 7536-7539 (1992). M. S. Rzchowski, Phys. Rev. B , 11745-11750 (1997). V. B. Cherepanov, I. V. Kolokolov, and E. V. Podivilov,JETP Letters , 596 (2001). S. E. Korshunov, Phys. Rev. B , 054416 (2002). M. R. Kolahchi and J. P. Straley, Phys. Rev. B , 144502(2002). M. J. Higgins, Yi Xiao, S. Bhattacharya, P. M. Chaikin, S.Sethuraman, R. Bojko and D. Spencer, Phys. Rev. B ,R894-R897 (2000). K. Park and D. A. Huse, Phys. Rev. B , 134522 (2001). Y. Xiao, D. A. Huse, P. M. Chaikin, M. J. Higgins, S. Bhat-tacharya, D. Spencer, Phys. Rev. B , 214503 (2002). R. J. Baxter, J. Math. Phys. , 784 (1970). V. B. Cherepanov, arXiv:cond-mat/9407068 (1994). S. Park and S. J. Lee, unpublished. K.-J. Koo, W.-B. Baek, B. Kim, and S. J. Lee, Jour. Kor.Phys. Soc. , 1977 (2006). A. S. Wills, V. Dupuis, E. Vincent, J. Hammann, and R.Calemczuk, Phys. Rev. B , R9264 (2000) (a)(b)FIG. 1: Schematic drawings of kagome lattices with the so-called (a) q = 0 ground state and (b) q = √ × √ . . . . . . . . T = 0 . (a) r C e q ( r ) . . . ∗ Tη (b) T η ( T ) . . . . . . . . . . . . . . . . FIG. 2: (a) Equilibrium spatial correlation of exp(3 iθ ) at various temperatures and (b) the corresponding exponents η ( T )vs. T , obtained from nonequilibrium relxation method. In (b), the dotted line represent 3 . T which fits reasonably well thebehavior of η ( T ) vs. T especially at low and intermediate temperature regime ( T ≤ . t = 20 (a) T = 0 . r C ( r , t ) . . . . t = 20 T = 0 . η = 0 . r r η C ( r , t ) . . . . ( c ) T = 0 . r/L ( t ) r η C ( r , t ) . . . . . . . . . . . . − . . . . . . . . . . . T = 0 . (d) t L ( t ) . . . . . . . . . . T = 0 . C = 0 . t β ( t ) . . . . . . . . . . . . . . . . T = 0 . C = 0 . t β ( t ) . . . . . . FIG. 3: (a) Equal-time spatial correlation function C ( r, t ) for exp(3 iθ ) for different time instants at T = 0 .
06, (b) the rescaledfunctions ˜ C ( r, t ) ≡ r η ( T ) C ( r, t ), (c) a scaling attempt based on (b) with C = 0 .
2, (d) growth of the length scale (using C = 0 . C = 0 . C = 0 . T = 0 . ξ v ξ h ξ t t ξ v , ξ h , ξ t FIG. 5: The length scales ξ h , ξ t and ξ v derived from the vortex densities versus time t for hexagonal, triangular plaquettes andthe sum of these, respectively at T = 0 . t n h (a) T = 0 . t n t , n h − − − . . . . . . T = 0 . (b) t n v ( = n t + n h ) − − = 0 . T = 0 .
07 (c) t n t − − − . . . . . . T = 0 . (d) t n h − − ( t )0 . ξ v (a) C = 0 . T = 0 . t ξ v , L ( t ) . L ξ v aL . ξ v (b) C = 0 . T = 0 . t ξ v , a L ( t ) . ( t )0 . ξ v (c) C = 0 . T = 0 . t ξ v , L ( t ) . L ξ v a ′ L . ξ v (d) C = 0 . T = 0 . t ξ v , a ′ L ( t ) . =0.0610mcs (a) =0.061280mcs (b) =0.0620480mcs (c) =0.06163840mcs (d) = 0 . T = 0 . t A ( t ) − − − − . . . . . . T = 0 . (a) t A ( t ) . . . . . T = 0 . (b)( log e ( t )) l o g e ( A ( t )) − − . − − . − − . −4