Yukawa particles confined in a channel and subject to a periodic potential: ground state and normal modes
J. C. N. Carvalho, W. P. Ferreira, G. A. Farias, F. M. Peeters
aa r X i v : . [ c ond - m a t . s o f t ] J un Yukawa particles confined in a channel and subject to a periodic potential: groundstate and normal modes
J. C. N. Carvalho, ∗ W. P. Ferreira, † G. A. Farias, ‡ and F. M. Peeters
1, 2, § Departamento de F´ısica, Universidade Federal do Cear´a,Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear´a, Brazil Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Dated: November 17, 2018)We consider a classical system of two-dimensional (2D) charged particles, which interact througha repulsive Yukawa potential exp ( − r/λ ) /r , confined in a parabolic channel which limits the motionof the particles in the y -direction. Along the x -direction, the particles are also subject to a periodicpotential substrate. The ground state configurations and the normal mode spectra of the systemare obtained as function of the periodicity and strength of the periodic potential ( V ), and density.An interesting set of tunable ground state configurations are found, with first and second orderstructural transitions between them. A magic configuration with particles aligned in each minimumof the periodic potential is obtained for V larger than some critical value which has a power lawdependence on the density. The phonon spectrum of different configurations were also calculated.A localization of the modes into a small frequency interval is observed for a sufficient strength ofthe periodic potential. A tunable band-gap is found as a function of V . This model system can beviewed as a generalization of the Frenkel and Kontorova model. PACS numbers: 64.60.Cn, 82.70.Dd, 63.20.D-
I. INTRODUCTION
Two-dimensional system (2D) are often created in thepresence of a substrate [1], which may induce a periodicpotential on the particles. In the pioneering experimentalwork of Chowdhury et al. [2], the authors studied a 2Dcolloidal system under influence of an one-dimensional(1D) periodic potential. An optical tweezer was used totrap the colloids by laser beams. For very high values oflight intensity, crystallization of the colloidal suspensionwas observed, when the periodicity of the substrate (pe-riodic potential) was chosen to be commensurate to themean particle distance. Laser induced freezing which iscaused by the suppression of thermal fluctuations trans-verse to the 1D periodic substrate was found (liquid-solidtransition) [3]. The system studied in Ref. [2] is relatedto the colloidal molecular crystal (CMC) and receivedattention recently due to important applications in pho-tonic and phononic crystals [4, 5].Specifically, CMC occurs when the number of colloidsis an integer multiple of the number of substrate min-ima, and has been investigated using in simulations [5, 6]and realized experimentally [7]. CMC is an interestingexperimental system to study order and dynamics in 2Dsince typical particle size and relaxation times permit,e.g, to use digital video-microscopy to track particle tra-jectories, allowing a deeper investigation of the physicalbehavior of the system [8]. ∗ Electronic address: joaoclaudio@fisica.ufc.br † Electronic address: wandemberg@fisica.ufc.br ‡ Electronic address: gil@fisica.ufc.br § Electronic address: [email protected]
Originally, CMC was proposed for 2D system in 2D pe-riodic potential (substrate). As known the dimensional-ity of the system plays an important role in many physicalproperties of distinct physical phenomena. In this sense,an interesting question is how the ordered structures andphysical properties would be influenced by the dimen-sionality of the periodic substrate. Recently, Herrera-Velarde and Priego [9, 10] studied a 2D system of re-pulsive colloidal particles confined in a narrow channeland subject to an external 1D periodic potential, whichcould be seen as the 1D version of the CMC. The mainfocus of the study was the role of the substrate in themechanisms that lead to a variety of commensurate andnon-commensurate phases, its effect on the the single-filediffusion regime, and the pinning-depinning transition in1D systems. The 1D character of the channel was rep-resented by a hard wall potential. Due to the repulsiveinteraction between the particles and the nature of theconfinement potential the density across the channel wasfound to be non-uniform with a higher density at thechannel edges.In the present paper we study the ordered configu-rations and the phonon spectrum of a 2D system ofrepulsive (Yukawa interaction) particles confined in aparabolic channel and subjected to a 1D periodic po-tential along the channel. As compared to the systemsin Refs. [9, 10], and due to the parabolic shape of theconfinement an opposite density distribution is observed,with particles more concentrated at the central region ofthe channel. As shown previously for finite size clustersof repulsive particles, the confinement potential is deter-minant, e.g. for melting and phonon spectra [11]. The1D parabolic confinement introduces a quasi -1D (Q1D)character to the system in the sense that particles are stillallowed to move freely in the perpendicular direction ofthe confinement potential.The interplay between the repulsive inter-particle in-teraction and the periodic potential determines the dif-ferent ground state configurations. Our model system ofYukawa particles can be realized experimentally using:i) a dusty plasma [12–14], ii) colloidal systems [15, 16]and iii) electrons on liquid helium [17, 18]. A dustyplasma consists of interacting microscopic dust particlesimmersed in an electron-ion plasma. The dust particlesacquire a net charge and the Coulomb interaction be-tween the dust-particles is shielded by the electron-ionplasma resulting in a Yukawa or screened Coulomb inter-particle interaction. The dust particles are confined to atwo-dimensional layer through a combination of gravita-tional and electrical forces. By microstructuring a chan-nel in the bottom electrode of the discharge it is possibleto laterally confine the dust particles as was realized inRefs. [19–23], the strength of the 1D confinement po-tential can be varied by the width of the channel or thepotential on the bottom electrode. When the width ofthe channel is microstructured into an oscillating func-tion along the channel it will result in a periodic potentialalong the channel.Alternatively, one can confine charged colloids, thatmove in a liquid environment containing counterions, intomicrochannels as e.g. recently realized experimentally in[24]. In this case the inter-colloid interaction can be mod-eled by a screened Coulomb interaction and the confine-ment potential is a hard wall potential. By changing thedepth profile of the micro-channel it has been shown in[25] that the confinement potential can be tuned intoa harmonic potential. Micro-structuring the width ofthe channel into an oscillating function along the channelwill result into an additional periodic potential along thechannel.In a previous work [26] the ordered configurations ofYukawa particles confined to Q1D were studied. A phasediagram was obtained as function of the particle densityand inverse Debye screening length which is a measure ofthe strength of the inter-particle interaction. The compe-tition between the lateral confinement and the screenedCoulomb interaction resulted in different phases wherethe particles are ordered in chains. The most well studiedphases are the one- and two-chain configurations wherethe transition between those two phases occurs through azig-zag transition. The latter is a continuous transition asfound theoretically for mono- [27] and bi-disperse [28, 29]systems, and experimentally [22, 23] with a power law de-pendence on the width [27, 30]. Here we are interestedto investigate how the phase diagram will be modifiedwhen an additional 1D periodic potential is present. Forexample, how the zig-zag transition will be modified bythe periodic potential.The present paper is organized as follows. In Sec. IIwe describe the model system and methods used in thecalculation of the properties. In Sec. III we present theresults for the different ground state configurations. InSec. IV the normal mode spectra for the one and two- chain regimes are presented for different intensities of theperiodic potential. Our conclusions are given in Sec. V.
II. THE MODEL
Our system consists of identical point-like particles in-teracting through a screened Coulomb potential. Theparticles are allowed to move in a two-dimensional (2D)plane and are subject to an external parabolic confine-ment in the y-direction and a periodic substrate potentialalong the x -direction. A sketch of the present model sys-tem is shown in Fig. 1. The total interaction energy ofthe system is given byH ′ = q ε X i The minimum energy configurations are obtained bynumerical and analytical calculations. In the numericalsimulations, we typically considered 100-200 particles, to-gether with periodic boundary conditions in the uncon-fined direction in order to mimic an infinite system. Wedo not consider friction in the present paper. In spite ofthe primary importance of friction to the motion of theparticles in real systems, the ground state configurationsare not affected by it.Notice that the substrate is defined in terms of theparameter L . Comparing L and a , we define here aninitially commensurate (IC) when ( L/a = p/q , with p and q integers) and initially non-commensurate (INC)regime of the ordered structures when the ratio L/a isa irrational number. It should be emphasized that inthese cases the inter-particle distance a is defined in theabsence of a substrate ( V = 0). In the case V = 0,it is expected that the mean distance between particlesalong a given chain a changes as a function of V , drivingthe system to new commensurate or non-commensurateconfigurations. III. GROUND STATE CONFIGURATIONS In this section, we present the results obtained ana-lytically and numerically for the ground state configura-tions. In the former, we calculate the energy per particlefor different configurations as a function of the strengthand periodicity of the substrate. We minimize such ex-pressions with respect to the different distances betweenparticles. The configuration with lowest energy is theground state. In order to predict which structures should be taken into account in the analytical approach, we alsouse molecular dynamic simulations as a complementarytool. The numerical method can give us some hints aboutwhich structures to consider. It should be noticed thatone of the draw backs of the numerical technique is thatin some cases there exist a larger number of meta-stablestates, mainly in the limit of high densities where thesystem is found in a multi-chain structure. On the otherhand, the numerical approach is the only way to obtainthe ground state configurations in some incommensurateregimes, which will be analyzed in the next sections.We show here that depending on the periodicity of thesubstrate we can tune the ground state configuration, in-duce structural phase transitions and control the numberof chains. This is interesting from an experimental pointof view, since the number of chains can be associatedwith the porosity of the system, making it a controllablefilter.The main features of the present model system can bealready seen in the more simple situations of the single-and two-chain regimes. For this reason, we limit our-selves to these cases, because it simplifies the physicalinterpretation of our results. A. Single-chain regime As an example, we study in this section systems withdensities n = 0 . n = 0 . 7, which are found in thesingle-chain regime [26] in the absence of the substrate V = 0. For n = 0 . L/a = 1 and L/a = 2, while for n = √ / L/a = √ n = 0 . L/a = 1. In thiscase, the configuration remains the same for any value of V . Notice that the cases in which L/a = 1 /I , where I ≥ L/a = I is very different and the particle configuration dependsstrongly on V , as will be shown in the next paragraph.In the IC case with n = 0 . L/a = 2, for smallvalues of V the particles are alternately located at theminimum and at the maximum of the substrate potential[see inset (a) in Fig. 2]. A sufficient increase of V forcesthe system to a new single-chain configuration in whicha pair of particles is located at each minimum of thesubstrate [see inset (b) in Fig. 2]. A further increaseof V pushes each pair of particles closer to each other,increasing the repulsive energy between them.For a critical value of V ( ≈ . 8) a structural transi-tion to the two-chain configuration is induced [see inset(c) of Fig. 2]. The system changes from a one to two-chains configuration. In the two-chain configuration theseparation d x between particles in the x-direction of eachminimum of the substrate is zero, which means particlesare aligned along the y-direction. The separation d y be-tween chains does not change as function of V . In thisparticular configuration d y is ruled only by the compe-tition between the repulsive interaction between parti-cles and the parabolic confinement, being independentof the strength of the periodic potential. The type oftransition observed here is different from the one foundin Ref. [27], where the authors demonstrated that in theabsence of a periodic potential and in the presence ofa parabolic confinement we have only continuous tran-sitions from one to two chains. In our case the one- totwo-chains transition is clearly a first-order phase tran-sition as function of V . Notice that in the two-chainregime the system is re-organized in a final commensu-rate structure with a new ratio L/a = 1. We can definehere a commensurate-commensurate transition betweendifferent orders of commensurability. FIG. 2: (Color online) Nearest neighbor separation betweenparticles in the x - ( d x ) and y -direction ( d y ) as a functionof V for the case n = 0 . L/a = 2. Three possibleconfigurations are shown as insets. The expression for the energy per particle which is ableto describe all phases observed in the case with n = 0 . L/a = 2 and in the case n = 1 . L/a = 1 isgiven by E = n X j e − κj/n j + n X j e − κn √ [( j − c x ] + c y q [( j − 1) + c x ] + c y + n X j e − κn √ ( j − c x ) + c y q ( j − c x ) + c y + 4 (cid:16) c y n (cid:17) − cos( πc x )(3)where c x = d x /L and c y = d y /L are, respectively, the di-mensionless separations between particles within a min-imum of the periodic potential along the chain and per-pendicular to it. FIG. 3: Ground state configurations for the case n = √ L/a = √ V = 0 . 17 (b) V = 0 . . Now we discuss the INC regime with n = √ L/a = √ 2. The same general behavior of previous casescan be observed here, with several structural transitionsruled by the strength of the periodic substrate V (Fig.3). For a large enough V the system can be found in afinal commensurate regime with L/a ≈ / 2, but now inthe three-chain configuration with particles almost uni-formly distributed over chains. B. Two-chain regime In this section we consider the system with n = 1 . V = 0. Differently from what was observedin the one-chain configuration ( n = 0 . L/a =1 . 0, the two chain configuration remains, but the internalstructure depends on V . This is shown in Fig. 4(a),where the relevant internal distances [Fig. 4(b)] for thearrangement are presented as function of V .For V = 0 . 16 the system changes from a staggered( d x = 0) to an aligned ( d x = 0) two-chain configu-ration, through a second order (continuous) structuraltransition, characterized by a discontinuity in the secondderivative of the energy with respect to V . Notice thatin the case L/a = 1 . 0, there are always two particles perperiod of the substrate potential, and in such a commen-surate phase the system is always found in the two-chainregime. FIG. 4: (Color online) (a) Inter-particle separation as func-tion of V for n = 1 . L/a = 1 . 0. (b) A sketch of thetwo-chain configuration with the distances d x and d y indi-cated. Next, we consider the more interesting case with n =1 . L/a = 2. When V is increased some unusualconfigurations appear as shown in Fig. 5. Initially theparticles move in the x-direction towards the minima ofthe periodic potential and at the same time each chainstarts to break up into two chains [Figs. 5(c)]. Thetransition found here is second-order.With further increase of V the two inner chains movetowards each other [see Fig. 5(d)] and merge into a sin-gle chain in the center [see Figs. 5 (d,e,f)]. The particlesin the outer chains move towards the minimum of theperiodic potential [see Figs. 5 (d,e,f)]. With further in-crease of V the pair of particles in the middle chain arepushed closer to each other and finally form a row of fourparticles directed along the y-direction and positioned inthe minimum of the periodic potential [see Figs. 5 (g,h)].The configurations presented in Fig. 5 indicates a tun-able porosity of the system as function of V . This is veryconvenient feature if the system is settled to be used asa filter or sieve, as pointed out in Refs. [32, 33], wheresuperparamagnetic colloidal particles were self-assembledin chain-like structures and used for separation of DNAmolecules. FIG. 5: Ground state configurations for the case n = 1 . L/a = 2 for different values of V . The movement of the different particles in the x- and y-direction as function of V is summarized in Fig. 6, wherestructural transitions are indicated by vertical dashedlines. Three second order structural transitions are ob-served as function of V and the number of chains variesfrom 2 → → → → n = 1 . L/a =1 . 5. Here the commensurate ratio changes according to V . Initially, for V = 0, the system is arranged in twochains [see Fig. 7(a)], which are displaced with respectto each other over half the inter-particle distance in eachchain. There are two particles per unit cell, which char-acterize an initially commensurate (IC) configuration.When V increases the system transits to a four-chainconfiguration through a second or first order structuraltransition, with the outer chains having twice as manyparticles as the inner chains [Fig. 7(b)]. Alternatively, wecan also view this configuration as two chains of trianglesas indicated in the shadowed region in Fig. 7(b). In thiscase d > d and d > d and the length of the unit cellis d = d + d . There are six particles in the unit cell, FIG. 6: (Color Online) (a) The lateral y -position of chainsfor the case n = 1 . L/a = 2. The vertical dotted linesrepresents the values of V where structural transitions occur.(b) The particle position in the x -direction as function of V . as in the case V = 0.With further increase of V , the y-distance d betweenthe internal chains goes to zero and the system changes tothe three-chain configuration [Fig. 7(c)] with the samenumber of particles in each one, and the central chainshifted by a/ V > . 5, particlesin different chains are all aligned along the y-directionand located in each minimum of the periodic substrate[Fig. 7(d)]. The trajectories of the different particles inthe channel as function of V is visualized in Fig. 8.Again, the relation between the periodicity of the sub-strate and the distance between particles is different from FIG. 7: (Color online) Ground state configurations for dif-ferent values of V for n = 1 . L/a = 1 . 5. In (b) therelevant distances for the analytical calculation of the energyare presented. the case V = 0, L/a = 1. This is interesting since wecan change the commensurability of the system by chang- FIG. 8: (Color Online) Particle trajectories for different val-ues of V near the potential minimum for the case n = 1 . L/a = 1 . ing only the strength of the substrate potential.As presented in Figs. 2(c), 4 and 5(h), for a criticalvalue of V the present model system is found in the spe-cial configuration where the particles are aligned alongthe confinement direction. Such a y -aligned configura-tion (YAC) occurs if the condition L/a = p , where p is an integer ( ≥ V = 0), thenwe find that the number of particles aligned along the y -direction in each minimum of the substrate potentialis N.p , which is also the number of chains. The criticalvalue of V for which the YAC phase can be induced isobtained by adding the interaction between particles andconfinement energies. A general expression for the YACis given by: V = nN p X j e − kNpj/n j + 2 nN p Np − X q =1 Np X l = q +1 X j e − kNp/n √ j +( p − q ) c p j + ( p − q ) c + nN p Np − X q =1 Np X l = q +1 e − k ( l − q ) cNp/n ( l − q ) c + 2 c N pn Np/ X l =1 ( l − / , (4)in the case where N.p is even, and FIG. 9: (Color online) (a) Critical value of V as function ofdensity for the YAC phase. (b) The distance between the par-ticles along the y-direction as function of the density in theYAC phase. The inset shows a sketch of the general ground-state configuration with all relevant parameters. The red linein both figures is the linear fit. V = nN p X j e − kNpj/n j + 2 nN p Np − X q =1 Np X l = q +1 X j e − kNp/n √ j +( p − q ) c p j + ( p − q ) c + nN p Np − X q =1 Np X l = q +1 e − k ( l − q ) cNp/n ( l − q ) c + 2 c N pn Np/ X l =1 l , (5)if N.p is an odd number.The critical value of V and the separation d betweenparticles in each minimum are presented in Fig. 9. Asketch of the configuration in each minimum of the peri-odic substrate with all relevant parameters is also shownas insect in Fig.9(b). A power law dependence of V and d/L on the density is found. IV. PHONON SPECTRUM Next, we analyze the V -dependence of the normalmode spectrum. We follow the standard harmonic ap-proximation and take into account the periodicity of thesystem in the unconfined direction ( x -axis).The present model system is a strictly 2D system wherethe number of particle in the unit cell and the number ofdegrees of freedom per unit cell determines the numberof branches in the phonon spectrum. If l is the num-ber of particles per unit cell, there will be 2 l branches inthe phonon spectrum, from which half of those branchescorrespond to oscillations along the chain, i.e along the x axis we have longitudinal modes, while the others areassociated with vibrations along the confinement direc-tion ( y axis transverse modes). If the particles in the unitcell oscillate in-phase, the mode is dominantly acoustical,while the opposite out-of-phase oscillation corresponds toan optical mode. In general, a normal mode can be clas-sified in one of the following classes: longitudinal opti-cal (LO), longitudinal acoustical (LA), transverse optical(TO), or transverse acoustical (TA).In the harmonic approximation the normal modes areobtained by solving the system of equations( ω δ µν,ij − D µν,ij ) Q ν,j = 0 , (6)where Q ν,j is the displacement of particle j from its equi-librium position in the ν direction, µ and ν refer to thespatial coordinates x and y , δ µν,ij is the unit matrix and D µν,ij is the dynamical matrix, defined by D µν,ij = 1 m X u φ µ,ν ( u ) e − iuqa , (7)where u is an integer assigned to each unit cell. The forceconstants are given by φ µ,ν ( u ) = ∂ µ ∂ ν exp( − κ p ( x − x ′ ) + ( y − y ′ ) ) p ( x − x ′ ) + ( y − y ′ ) , (8)with ( x − x ′ ) = distance between particles along the x-axis and ( y − y ′ ) = interchain distance with ( x, y ) and( x ′ , y ′ ) the equilibrium positions of the particles in theunit cell, and φ µ,ν ( u = 0) = − X u =0 φ µ,ν ( u ) . (9)The phonon frequency is given in units of ω / √ 2. Asan example, the complete dynamical matrix for the one-chain and the two-chain regime are given in Appendix.The frequencies for the one-chain configuration in thecase V = 0, are given by ω l = √ A for the acousticalbranch and ω t = √ A for the optical branch, A and A are defined in Appendix. FIG. 10: (Color online) The phonon spectrum for differentvalues of V in the case n = 0 . L/a = 1. The frequencies for the one-chain configuration whenwe have V = 0 and two-chain configuration can be givenby: ω l = r 14 ( B + B ± q B + 4 B B − B B + B + sub )(10)for the longitudinal modes, and by ω t = 12 r B + B ± q B + 4 B B − B B + B (11)for the transverse modes. The expressions for B , B , ..., B and are given in Appendix. Here sub =8 V π cos ( πc x ) is the term related to the periodic sub-strate. The wave number k for the one- and the two-chains regimes is in units of 2 π/L , where L is the lengthof the unit cell in the x -direction.In Fig. 10(a), the phonon spectrum for the one-chainconfiguration is presented for different values of V , fixeddensity n = 0 . L/a = 1. In this case, there isone particle per unit cell located in each minimum of thesubstrate resulting only in one longitudinal mode andone transversal mode. The frequency of the longitudinalmode increases with increasing V , and there is a gapopening at k = 0. The reason is that the periodic po-tential acts locally as a parabolic confinement potential V ( x ) ≃ V π L x with frequency ω = p V /m πL . The k = 0 gap corresponds with this frequency for not toosmall values of V . The transversal mode correspondsto particle oscillations in the y-direction and is thereforepractically independent of V .In Fig. 11, the dispersion curves for n = 0 . L/a = 2 are presented for different values of V . As ob-served in Fig. 2, the presence of the substrate ( V = 0) FIG. 11: The phonon spectrum for different values of V inthe case n = 0 . L/a = 2. modifies the number of particles in the unit cell in or-der that the number of branches of the phonon spec-trum is increased as compared to the case V = 0 [Fig.11(a)]. For V = 0 there are two particles per unit celland consequently four branches in the phonon spectrum.As V increases, the frequency of the LA mode also in-creases, which can be explained keeping in mind that forlow values of V , there is a small electrostatic repulsionbetween neighboring particles, in order that particles os-cillate horizontally without major difficulties. The oppo-site behavior is found for the TO mode, i.e., decreaseswith increasing V . The distance between adjacent par-ticles in the same substrate minimum becomes smaller,and the repulsive force between them increases and actsas a retarding force.The LO mode has a rather different behavior as com-pared to the TO mode, i.e., there is a hardening of itsfrequency when V increases, which is a consequence ofthe larger repulsion due to the closer proximity betweenparticles. For a sufficiently strong V [ Fig. 11(c)] thenormal mode spectrum becomes discrete, i.e. frequen-cies become independent on k , which means the groupvelocity is zero and the modes become localized.As commented previously, for V ≥ . n = 1 . L/a = 1, [Fig. 12]. As pre-sented earlier [Fig. 4], particles remain in the two-chainsconfiguration for all V with changes only in the internalstructure. Again the substrate potential induces gaps inthe normal mode frequencies as presented in Fig. 12.The TA, TO, LA and LO modes increase with increasing V .In the case of the LA mode, for low values of V par-ticles are not aligned, having more freedom to oscillatein the horizontal direction. When V increases, the elec-trostatic force becomes larger (particles are now aligned)making oscillations along the channel harder.The LO mode also increases with increasing V . This isa consequence of the strength of the substrate potential,which trap particles in their equilibrium positions, reduc-ing the out phase oscillations of the particles. The TOfrequency increases slightly, since the out-of-phase mo-tion is more difficult to occur. The TA frequency branchis almost independent of V because it corresponds to os-cillations in the y -direction and is therefore determinedby the harmonic confinement potential with frequency ω .For the YAC phase, V > . 16, the normal modespectrum becomes discrete [Fig. 12(d)], as in the case n = 0 . L/a = 2 . 0. The modes are almost constant0 FIG. 12: The phonon spectrum for different values of V inthe case n = 1 and L/a = 1. due to the strong confinement potential imposed by thesubstrate and the harmonic trap. V. CONCLUSIONS We investigated the structural and dynamical prop-erties of a two-dimensional system of repulsive particlesconfined by a parabolic channel and submitted to a one-dimensional periodic potential (substrate). The groundstate configurations were obtained analytically and nu-merically, where for the latter we used molecular dynam-ics simulations. The phonon spectrum were also calcu-lated analytically for the one- and two-chain configura-tions through the harmonic approximation.The main features of the structure and normal modespectrum were studied (for different densities) as a func-tion of the periodicity ( L ) and strength ( V ) of the sub-strate, which are experimentally tunable parameters insystems like e.g. colloids in the presence of a periodiclight field composed of two interfering laser beams. Aninteresting set of ground state configurations with con-trollable porosity is observed mainly as a function of V ,through several first or second order structural transi-tions. The structures are mainly ruled by the fact thatparticles tend to go to the minima of the periodic sub-strate, modifying the symmetry of the ordered structures.However, for small V the inter particle repulsive interac-tion dominates and the particles are found over all possi-ble positions in the periodic potential, including regionsnear to the maxima. For large V , particles are more andmore attracted to the wells of the periodic potential.For some specific cases we found structural transitionswhere the number of particles in the unit cell of the peri-odic system is changed, implying e.g. a different numberof branches in the phonon spectrum, which is an inter-esting aspect of the dynamical behavior of the system,specially for applications in phononics.The normal mode frequencies depend on the lineardensity of the system, periodicity and strength of the pe-riodic substrate. We observed gaps in the phonon spec-trum, which indicate that there are frequencies blockedby the crystal. For V beyond a critical value and for spe-cific values of the ratio L/a the system is found in a spe-cial configuration were particles are aligned in each min-imum of the periodic substrate and perpendicular to the x -direction. For such a configuration the normal modefrequencies become independent of the wave vector andthe modes localize into a small frequency of interval. Acknowledgments JCNC, WPF, GAF and FMP were supported bythe Brazilian National Research Councils: CNPq andCAPES and the Ministry of Planning (FINEP). FMPwas also supported by the Flemish Science Foundation(FWO-Vl).1 Appendix The matrix ω I − D (where I is the unit matrix and D is the dynamical matrix) is used in the calculation of thenormal modes for the one- and two-chains configurations.The dynamical matrix for one chain configuration when V = 0 is: (cid:20) ω − A 00 ( ω − ω ) − A (cid:21) , where the quantities A and A are given by: A = ∞ X j =1 n e − κj/n j h κjn + κ j n i [1 − cos ( kπj )]+ V πn cos ( πj ) (A.1) A = ∞ X j =1 n e − κj/n j h κjn i [1 − cos ( kπj )] (A.2)The dimensionless wave number k is in units of 2 π/L .The dynamical matrix to one-chain V = 0 and two-chains configuration is: ω − B − sub − B 00 ∆ ω − B − B − B ω − B − sub − B ω − B , where ∆ ω = ω − ω . The quantities B , B , B , B , B and B are given by: B = ∞ X j =1 n e − κr/n (2 r ) h ( j − c x ) (cid:16) r + 6 κnr + 4 κ n (cid:17) − (1 + 2 κrn ) i + ∞ X j =1 n e − κj/n (2 j ) h κjn + (2 κj ) n i × [1 − exp ( ikjL )] (A.3) B = ∞ X j =1 n e − κr/n (2 r ) h c y r + 4 κ c y n + 6 κc y nr − (1 + κrn ) i − ∞ X j =1 n e − κj/n (2 j ) h κjn i [1 − exp ( ikjL )] , (A.4) B = ∞ X j =1 n e − κr /n (2 r ) h ( j − c x ) (cid:16) r + 6 κnr + 4 κ n (cid:17) − (1 + 2 κr n ) i + ∞ X j =1 n e − κj/n (2 j ) h κjn + (2 κj ) n i × [1 − exp ( ikjL )] (A.5) B = ∞ X j =1 n e − κr /n (2 r ) h c y r + 4 κ c y n + 6 κc y nr − (1 + κr n ) i − ∞ X j =1 n e − κj/n (2 j ) h κjn i [1 − exp ( ikjL )] (A.6) B = ∞ X j =1 n e − κr/n (2 r ) h ( j − c x ) (cid:16) r + 6 κnr + 4 κ n (cid:17) − (1 + 2 κrn ) i [ exp ( ikL ( j − c x ))] (A.7) B = ∞ X j =1 n e − κr/n (2 r ) h c y r + 4 κ c y n + 6 κc y nr − (1 + κrn ) i × [ exp ( ikL ( j − c x ))] , (A.8) B = ∞ X j =1 n e − κr /n (2 r ) h ( j − c x ) (cid:16) r + 6 κnr + 4 κ n (cid:17) − (1 + 2 κr n ) i [ exp ( ikL ( j + c x ))] (A.9) B = ∞ X j =1 n e − κr /n (2 r ) h c y r + 4 κ c y n + 6 κc y nr − (1 + κr n ) i × [ exp ( ikL ( j + c x ))] , (A.10)where r = q ( i − c x ) + c y , r = q ( i − c x ) + c y ,the dimensionless wave number k is in units of 2 π/L , i = √− sub = 8 V π cos ( πc x ).2 [1] H. 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