2D mobile breather scattering in a hexagonal crystal lattice
22D mobile breather scattering in a hexagonal crystal lattice
J. Bajars (1) , J. C. Eilbeck (2 , , and B. Leimkuhler (2 , Faculty of Physics, Mathematics and Optometry, University of Latvia,Jelgavas Street 3, Riga, LV-1004, Latvia (2)
Maxwell Institute, (3)
School of Mathematics, University of EdinburghJames Clerk Maxwell Building, The King’s Buildings,Mayfield Road, Edinburgh EH9 3JZ, UK, (4)
Department of Mathematics, Heriot-Watt University,Riccarton, Edinburgh EH14 4AS, UK (Dated: July 24, 2020)
Abstract
We describe, for the first time, the full 2D scatteringof long-lived breathers in a model hexagonal lattice ofatoms. The chosen system, representing an idealizedmodel of mica, combines a Lennard-Jones interatomicpotential with an “egg-box” harmonic potential wellsurface. We investigate the dependence of breatherproperties on the ratio of the well depths associatedto the interaction and on-site potentials. High valuesof this ratio lead to large spatial displacements inadjacent chains of atoms and thus enhance the twodimensional character of the quasi-one-dimensionalbreather solutions. This effect is further investigatedduring breather-breather collisions by following theconstrained energy density function in time for a set ofrandomly excited mobile breather solutions. Certaincollisions lead to 60 ◦ scattering, and collisions of mobileand stationary breathers can generate a rich variety ofstates.The nature of mysterious particle-like tracks in mus-covite mica crystals have attracted much recent interest[1]. The lines were first observed by Russell over 50 yearsago [2], who suggested that they were caused by local-ized vibrational modes (which he called quodons) in theK-K layer of mica. This hypothesis has lead to a numberof simulations of breathers in model hexagonal latticeswith on-site potentials [3–6]. The surprising conclusionof these studies is that in similar models in 2D, local-ized single breathers can travel along the main crystaldirections of the lattice with little attenuation or lateralspreading.In this note we move beyond the case of singlebreathers by examining breather-breather collisions. Wepresent evidence that breathers are remarkably robust tocollisions, and scattering through some multiple of 60 ◦ into another crystal direction is frequently observed insome circumstances. In addition we examine ensemblesof initial conditions for breather-breather collisions to be-gin to understand how the relative angles and phases ofthe breathers affect their interactions. Our simplified 2D model of the hexagonal K-K sheetlayer in mica crystal [6] is based on the following di-mensionless Hamiltonian which describes the classical dy-namics of N potassium atoms: H = K E + U + V c (1)= N (cid:88) n =1 (cid:18) | ˙ r n | + U ( r n ) + 12 N (cid:88) n (cid:48) =1 n (cid:48)(cid:54) = n V c ( | r n − r n (cid:48) | ) (cid:19) , where K E is the kinetic energy, U is the on-site potentialenergy (modelling forces from atoms above and belowthe K-K sheet), V c is the radial interparticle potential ofthe potassium atoms with a cut-off radius r c , r n ∈ R isthe 2D position vector of the n th K atom, ˙ r n is its timederivative, and | · | is the Euclidean distance. Note thatno motion in the z -direction is allowed. Any mentionof “transverse” in the following means in-plane motiontransverse to the breather propagation line.The dimensionless on-site potential U is modelled as asmooth periodic function resembling an egg-box cartonwith hexagonal symmetry: U ( x, y ) = (cid:18) − (cid:18) cos (cid:18) π ( √ x − y ) √ (cid:19) (2)+ cos (cid:18) π ( √ x + y ) √ (cid:19) + cos (cid:18) πy √ (cid:19)(cid:19) , where x and y are configurational coordinates, r n =( x, y ). Importantly, in any of the three crystallographiclattice directions with direction cosine vectors (1 , T and(1 / , ±√ / T , the on-site potential (2) is a cosine, so themodel reduces to a special case of the Frenkel-Kontorovamodel. The 1D atomic chains in the (1 , T lattice di-rection are denoted by y m , where m ∈ Z . The inter-atomic interactions of K atoms are modelled by a scaledLennard-Jones potential V LJ ( r ) with cut-off radius r c ,i.e. V c ( r ) = (cid:15) (cid:32)(cid:18) r (cid:19) − (cid:18) r (cid:19) (cid:33) + (cid:15) (cid:88) j =0 A j (cid:18) rr c (cid:19) j , (3) a r X i v : . [ n li n . PS ] J u l if 0 < r ≤ r c , and zero elsewhere. The cut-off dimension-less coefficients A j → r c → ∞ are determinedfrom matching and continuity conditions on V LJ at welldepth r = 1 and the cutoff r c , respectively, see [6] formore details.In this paper we consider r c = 3 in dimensionless units.In general, we did not observe qualitatively different re-sults for r c ≥ √
3. This can be attributed to the rapiddecay of the Lennard-Jones potential (3) for r → ∞ .The dimensionless parameter (cid:15) controls the relativestrength of the two potentials. If (cid:15) = 0 then the systemdecouples into nonlinear oscillators whereas, for (cid:15) → ∞ ,the system behaves as a Lennard-Jones fluid. Previousstudies [3] suggest that propagating breather solutionsare observed when the two potentials are of roughly equalrelative strength. For the system (1), mobile breather so-lutions can be observed [6] for (cid:15) ∈ [0 . , without an on-site potential, see for example [7, 8] and referencestherein. Ref. [8] discusses general cases of the so-called“crowdions”, a modern name given to pulses called kinksin the older literature. Ref. [7] discusses solitons colli-sions with 2D scattering but the scattered pulses are notlong-lived. Breather collisions are discussed in [9], butno cases involving 60 ◦ are described.To excite mobile discrete breathers, the simplestmethod is to consider atoms in their dynamical equi-librium state, i.e. at the bottom of each well of the on-site potential (2), and excite three co-linear neighbouringatomic momenta in any crystallographic lattice directionwith the pattern v = γ ( − , , − T , (4)where the values of γ ∈ R depend on the choice of (cid:15) . Incontrast to other initial excitations such as single kicksor more complex patterns, we find that this pattern pro-duces clean initial conditions for the excitation of mo-bile discrete breather solutions, i.e. produced very fewphonons. Similarly, by considering patterns involvingfour co-linear atomic momenta w = γ ( − , , − , T ,we are able to excite stationary breathers. In the presentstudy we concentrate on breather-breather interactionsand therefore avoid the complications that a higherphonon density would bring.We integrate the Hamiltonian dynamics of the systemwith the second order time reversible symplectic Verletmethod [10, 11]. In the following, all numerical exam-ples are performed with time step τ = 0 .
04 and peri-odic boundary conditions for different values of (cid:15) and γ .We can define an energy density function by assigning toeach atom its kinetic energy and on-site potential valuesas well as half of the interaction potential values. To ob-tain positive values we redefine H := H + | min { H }| suchthat H ≥
0. In all energy density plots we interpolate H over a square uniform mesh. The initial excitation (4) leads [6] to highly local-ized mobile breather solutions propagating on a chainof atoms in a crystallographic lattice direction with largedisplacements in the x direction, almost zero displace-ments in the y -axis direction and with small displace-ments in both axis directions on the chains of atomsadjacent to the main chain of atoms. In addition, theobserved mobile breathers are optical with internal fre-quencies above the phonon linear spectrum.In considering breather collisions, there are three pos-sibilities. The first is inline or head-to-head collisionswith two breathers on the same chain but travelling inopposite directions. These were first looked at brieflyin [4]. The second occurs when two breathers approacheach other along the same lattice vectors but on adjacentparallel chains. The third occurs when two breathers ap-proach along different lattice vectors, i.e. at an angle ofa multiple of 60 ◦ .We discuss in detail elsewhere how the choice of (cid:15) and γ affects the breather properties. In general increasing γ generates a narrower and faster moving breather in thedirection of travel, but has little effect on the width ofthe breather perpendicular to the direction of travel. In-creasing (cid:15) , which also increases the speed, in contrastto the γ variable, increases the maximal displacementsperpendicular to the direction of travel. This transversespreading is important when discussing breathers collid-ing on adjacent, parallel tracks. In the present paper forconciseness, we consider only the values (cid:15) = 0 . , . (cid:15) =0 .
05 and a head-to-head collision shown in Figure 1. Con-sider a periodic lattice of size N x = 400 and N y = 32 ofatoms in their equilibrium state and launch two atomicexcitations (4) in the middle of the lattice at each ends ofchain y m . We indicate left and right excitation parame-ter values by γ l and γ r , respectively of opposite signs, toset the breathers on a collision course. We set γ l = 0 . γ r = − .
5, and integrate in time until T end = 1200.At around t = 700, the breathers collide, pass throughand exchange some energy, see Figure 1.In Figure 1(a) we plot the energy density function intime of the atoms on the main chain y m where the great-est energy of the breather solution is localized. Noticethe slight change in the breather propagation speed af-ter the collision. This is also confirmed by the frequencyspectrum plot in Fig. 1(b). Note also the breather fre-quency focusing in time before and after the collision,as was observed in [6]. Figure 1(b) clearly shows thatthe breather solutions have exchanged their energies anddominant internal frequencies during the collision.The result of Figure 1 can be thought of as demon-strating a strongly one-dimensional nature, despite the2D nature of the lattice. Due to the chaotic nature ofmolecular dynamics, the numerical observations, partic-ularly at long times, are sensitive to changes in the initial (a)(b) FIG. 1. Simulation of two mobile breather head-to-head col-lision. (a) energy density function on the main atomic chainof breather propagation. (b) frequency spectrum of atomicdisplacement function ∆ x m ( t ) in x -axis direction from equi-librium. conditions and to round-off error. This motivates us toconsider an ensemble of initial conditions as well as dif-ferent starting configurations, to study breather-breathercollisions.For the ensemble, we draw two sets of normally dis-tributed random numbers X, Y ∼ N (0 ,
1) and scale themto normally distributed numbers with mean values γ l,r and variance σ l,r . Thus we obtain two sets of the ex-citation parameter value γ ∼ N ( γ l,r , σ l,r ) for numericalsimulations. For the following examples on the lattice N x = 200 and N y = 64, we consider two sets of 2000 ran-dom numbers sampling the standard Cauchy distribution(i.e. of the ratio X/Y ) and scale parameters equal to zeroand one, respectively, with mean values γ l,r = ± . σ l,r = 0 . (a)(b) FIG. 2. Constrained (
H > .
01) energy density function aver-aged over 50 time snapshots and 2000 individual simulationsof breather-breather collisions on adjacent chains of atoms.(a) simulation with (cid:15) = 0 .
01 until T end = 1000. (b) simula-tion with (cid:15) = 0 .
05 until T end = 500. present in the lattice, from phonons generated from theinitial excitations, we set atomic energy density valuesto zero if the value is smaller than 0 .
01. This value isestimated from the numerical observations. Thus mostof the phonon energy is disregarded for the final energyaverages and most of the information comes from thebreather solutions.Using this set of initial velocities for inline collisions(as in Fig. 1) we did not observe scattering of breathersolutions into different crystallographic lattice directionsdespite a visible spread of energy around the main chainof atoms of breather propagation. If instead we considerthe scattering of two breathers on adjacent parallel lines,the results depend on the value of (cid:15) used.For (cid:15) = 0 .
01, Fig. 2(a), we observed no scattering ofbreather solutions into different crystallographic latticedirections. However for (cid:15) = 0 .
05, we observe breatherscattering in all lattice directions, Fig. 2(b). Noticethat dark energy lines arise only in these directions in-dicating propagating as well as stationary breather so-lutions. These examples demonstrate the 2D properties
FIG. 3. Constrained (
H > .
01) energy density function aver-aged over 50 time snapshots and 2000 individual simulationsof mobile breather collisions with a stationary breather on acrystal axis at 60 ◦ to the moving one. The stationary breatheris at slightly different positions in each case. of breather solutions, breather energy scattering and theimportance of the parameter (cid:15) . Figure 2(b) confirms thatthe mobile breather’s 2D character increases for largervalues of (cid:15) , that is, for a stronger interaction potential rel-ative to a weaker on-site potential energy. Since Hamil-tonian (1) is time reversible, Fig. 2(b) also demonstratesbreather-breather collisions with an angle to each otherwhen the time is reversed.To explore further breather energy scattering bybreather-breather collisions in 2D lattice model (1) weconsider simulations of mobile breather collisions with a stationary breather at an angle to the incoming in Fig. 3.We consider four different locations of the (1 , − , , − N x = 120 and N y = 64, and integrate until T end = 500.We consider a mean excitation value of γ l = 0 . γ r = − .
35 where γ r refers to the stationary breatherand variances σ l = 0 .
001 and σ r = 0 . phases of the two breathers will also bedifferent in each simulation.As above, we observe stronger breather energy scatter-ing in the computations with (cid:15) = 0 .
05, Fig. 3, in contrastto the simulations with (cid:15) = 0 .
01, (not shown), where thescattering is predominately only in the lattice directionsof both breathers.Not only do we see scattering through a multiple of 60 ◦ ,but the details of which track the breathers scatters to is very sensitive to the velocity and phase of the incomingbreather as well as the position of the stationary breather.Our study has given us a better understanding ofparticle-like tracks in muscovite mica crystals. Wedemonstrate the importance of the relative strengths ofthe interatomic force and of the force from the surround-ing atoms for the existence of long-lived propagatingbreathers and their 2D collision properties. Recent ex-perimental work by Russell et al. [12] suggests stronglythat breather-like objects are important in real 3D crys-tals of several different materials, displaying hypercon-ductivity and annealing effects at finite temperatures de-spite a range of defects such as impurities, dislocations,and crystal boundaries. We plan to extend the currentmodel to one covering more realistic physical situations.JB, during his postdoctoral research at the Univer-sity of Edinburgh, and BJL acknowledge the support ofthe Engineering and Physical Sciences Research Coun-cil which has funded this work as part of the NumericalAlgorithms and Intelligent Software Centre under GrantEP/G036136/1. JCE thanks Mike Russell for many use-ful conversations. [1] J. F. R. Archilla et al. , eds., Quodons in mica: nonlin-ear localized travelling excitations in crystals , SpringerSeries in Materials Science, Vol. 221 (Springer Interna-tional Publishing, 2015).[2] F. M. Russell, Nature , 907 (1967).[3] J. L. Mar´ın, J. C. Eilbeck, and F. M. Russell, PhysicsLetters A , 225 (1998).[4] J. L. Mar´ın, J. C. Eilbeck, and F. M. Russell, in
Non-linear Science at the Dawn of the 21th Century , editedby P. L. Christiansen et al. (Springer, Berlin, 2000) pp.293–306.[5] J. Bajars, J. C. Eilbeck, and B. Leimkuhler, in [1] (2015)pp. 35–67.[6] J. Bajars, J. C. Eilbeck, and B. Leimkuhler, Physica D:Nonlinear Phenomena , 8 (2015).[7] A. Chetverikov, W. Ebeling, and M. G. Velarde, Letterson materials , 82 (2016).[8] I. A. Shepelev, E. A. Korznikova, D. V. Bachurin, A. S.Semenov, A. P. Chetverikov, and S. V. Dmitriev, PhysicsLetters A , 126032 (2020).[9] A. A. Kistanov, V. D. Sergey, A. P. Chetverikov, andM. G. Velarde, Eur. Phys. J. B , 211 (2014).[10] M. P. Allen and D. J. Tildesley, Computer Simulation ofLiquids , Oxford science publications (OUP, USA, 1989).[11] B. Leimkuhler and S. Reich,
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