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recently , there has been considerable theoretical and experimental progress in the study of mesoscopic systems consisting of a finite number of charged particles which are confined into an artificial circular symmetric potential . in 1934 , wigner suggested that a liquid to solid phase transition should occur in a three - dimensional ( 3d ) fermi system at low densities @xcite . typical experimental model systems for the study of this system are electrons on the surface of liquid helium @xcite , electrons in quantum dots @xcite , colloidal suspensions@xcite and in confined plasma crystals@xcite . on the other hand , various similar systems , like the vortex clusters in an isotropic superfluid @xcite , vortices in superfluid he@xmath0@xcite , vortices in a bose - einstein condensate stirred with a laser beam @xcite and in mesoscopic superconducting disks @xcite have many common features with those of 2d charged particles . colloidal particles dissolved in water @xcite are another example of an experimental system where classical particles exhibit wigner crystallization . recently , macroscopic 2d wigner islands , consisting of charged metallic balls above a plane conductor were studied and ground state , metastable states and saddle point configurations were found experimentally @xcite .    such a system with a finite number of particles , initially studied by thomson as a classical model for the atom @xcite , has been extensively studied during the past few years . for a small number of particles ( typically @xmath1 ) they are arranged in rings @xcite and a mendeleev - type of table was constructed in ref . @xcite which gives the distribution of those particles over the different rings . moreover , the configurations of the ground state , the metastable states and saddle point states were obtained , from which the transition path and the geometric properties of the energy landscape were given in ref . the spectral properties of the ground state configurations were presented in ref . @xcite and generalized  to screened coulomb @xcite and logarithmic @xcite interparticle interactions . the excitation of normal modes of 2d coulomb clusters in laboratory complex plasmas were recently observed @xcite . the melting properties of this system have been studied experimentally @xcite and by monte carlo(mc ) @xcite and molecular dynamics @xcite simulations . in a hard wall confined system with short - range inter - particle interaction , the melting behavior was found even more interesting . reentrant melting of 2d colloidal clusters in a hard wall potential was obtained in both experimental @xcite and theoretical work @xcite . the defect structure in crystals is of paramount importance for the stability and the strength of materials . topological defects in wigner crystals  @xcite and its effect on particle melting were investigated in refs . thermal defect mediated melting was proposed as the microscopic mechanism for melting in an infinite 2d triangular wigner crystal . it is well known that the kthny scenario(kosterlitz - thouless - halperin - nelson - young ) describes 2d melting as a defect - mediated phenomenon and melting occurs in two stages through two continuous phase transitions @xcite . a clear case of two - stage melting was observed in a film of paramagnetic colloidal particles @xcite . in this paper we study topological defects which are induced by the confinement potential , i.e. which are a result of the finite size of the system . next we investigate how these defects influence the melting of the mesoscopic 2d island . the present paper is organized as follows . in sec . ii , we describe the model system and the numerical approach . sec . iii is devoted to the structural properties of the topological defects at zero temperature . in sec . iv , we discuss the eigenmode spectrum for these large clusters . the discussion on the non - homogeneous melting is presented in sec . v. our conclusions are given in sec . the model system was defined in ref . @xcite and the hamiltonian for such a system is given by @xmath2 the confinement potential @xmath3 is taken circular symmetric and parabolic , where @xmath4 is the effective mass of the particles , @xmath5 is the particle charge , @xmath6 is the radial confinement frequency and @xmath7 is the dielectric constant of the medium the particles are moving in . to exhibit the scaling of the system , we introduce the characteristic scales in the problem : @xmath8 for the length , @xmath9 for the energy and @xmath10 for temperature . after the scaling transformations ( @xmath11 ) , the hamiltonian can be rewritten in a simple dimensionless form as    @xmath12    which only depends on the number of particles @xmath13 . the numerical values for the parameters @xmath14for some typical experimental systems were given in ref . @xcite . the mc simulation technique @xcite is relatively simple and rapidly convergent and it provides a reliable estimation of the total energy of the system in cases when relatively small number of metropolis steps is sufficient . however , the accuracy of this method in calculating the explicit states is poor for systems with a large number of particles , which have significantly more metastable states . to circumvent this problem we employed the newton optimization technique which was outlined and compared with the standard monte carlo technique in ref . the structure and potential energy of the system at @xmath15 @xmath16 are found by the standard metropolis algorithm @xcite in which at some temperature the next simulation state of the system is obtained by a random displacement of one of the particles . we allow the system to approach its equilibrium state at some temperature @xmath15 , after executing @xmath17 `` mc steps '' . each mc step is formed by a random displacement of all particles . if the new configuration has a smaller energy it is accepted , but if the new energy is larger the configuration is accepted with probability @xmath18 where @xmath19 is a random number between 0 and 1 and @xmath20 is the increment in the energy @xcite . it is well known that the hexagonal lattice is the most energetically favored structure for classical point charges in a two - dimensional infinite plane at low temperature @xcite . for a system consisting of a finite number of repelling particles restricted to 2d , which are held together by a circular harmonic potential , the cluster patterns are determined by the need to balance the tendency to form a triangular lattice against the formation of a compact circular shape . the configuration is determined by these two competing effects , namely circular symmetry and triangular structure ( wigner lattice ) . this competition leads to intrinsic defects in the 2d circular coulomb cluster which are _ geometry _ ( of the confinement potential ) _ induced defects_. this ground state is not a defect free system . the symmetry breaking is due to the packing of the triangular lattice into a region with a circular boundary . a hexagonal lattice which is cut by a circle without the introduction of any defect has an energy @xmath21 which is larger than the ground state energy @xmath22 for @xmath23 particles .    in the first part of this paper , we investigate the form and position of the defects in large clusters . therefore we make use of the voronoi construction @xcite . the voronoi construction of a collection of particles consists of a partition of space into cells . each cell consists of those points which are closer to the given particular particle than to any other particles . examples of voronoi constructions are shown in fig . [ fig1 ] where the ground state configuration for @xmath24 and @xmath25 are shown . one can see that there are two kinds of defects , i.e. dislocations and disclinations . disclinations are orientational defects with five ( indicated by `  ' ) or seven ( indicated by ` + ' ) fold coordination number ( the number of sides of the polygon around the particles is nothing else then the coordination number ) . a dislocation is a pair of two disclinations consisting of a defect with 5-fold (  ) and a defect with 7-fold ( + ) coordination number . in the latter case the ordering at long distances is not disrupted and consequently such a bound pair has a much lower energy @xcite . the total number of 5-fold @xmath26 and 7-fold @xmath27 disclinations depends on the particular configuration . the number of disclinations in this system is determined by euler s theorem and ca nt be changed , so the net topological charge @xmath28 is always equal to six as was already demonstrated in refs . the reason is that every `  ' defect can bend the lattice clockwise over @xmath29 from a straight lattice and thus six `  ' defects can bend a straight line into a circle . dislocations will appear when it decreases the energy of the system . from fig . 1 it is apparent that this is more so for larger clusters .    in refs . @xcite , the defects in clusters with a logarithmic inter - particle interaction were studied . we want to stress that their way of visualizing the defects is different : nearest neighbours are connected by a line , without making crossings . however , this does not lead to a unique picture : the total number of 5-fold , @xmath30 and 7-fold , @xmath31 disclinations can vary in the same configuration , only the net topological charge @xmath28 is always equal to six .    in these large clusters , the defects are located on a hexagon , i.e. they form a hexagonal structure . as can be seen in fig . [ fig1 ] the defects are approximately situated at the six corners of a hexagon , each corner with a net topological charge of ` 1 ' . notice from fig . [ fig1 ] that a single 5-fold disclination can appear , but never a single 7-fold disclination . for the large strain energy around ` 1 ' topological charge , some dipole defects(i.e . dislocation with ` 1 ' and ` + 1 ' defects ) will be generated to shield the ` 1 ' topological charge . those shielding dipole defects do not change the topology of the system . a clearer example is shown in fig . [ fig2 ] for the @xmath23 groundstate configuration . we considered the @xmath23 system as it minimizes the number of defects . the reason is that for this particle number the configuration has 42 particles in the outer ring , which is a multiple of the topological charge . there are three rings at the border with an equal number ( @xmath32 ) of particles ( the 1d wigner lattice ) , the central hexagonal structure ( the 2d wigner lattice ) and the defects indicated by triangles ( @xmath33 ) and squares ( @xmath34 ) are situated : ( i ) around the six corners of a hexagon and ( ii ) in the transition region between the outer rings and the central hexagon . it should be noted that the search for the global minimum configuration is a difficult problem for large systems because of the existence of a large number of local - minimum configurations , with energy very close to the global minimum . thus one is never @xmath35 sure to have found the real ground state . therefore , we investigated the different metastable states . in an experiment , those metastable states may be reached by thermal excitation if the energy barrier between them and the ground state is comparable to or smaller than @xmath36 . the saddle points between those metastable states were investigated in an earlier paper @xcite for @xmath37 . in fig . [ fig3 ] the energy and the total number of defects of different metastable states are shown for @xmath38 the results for the different metastable configurations are ordered with increasing energy . note from fig . [ fig3](b ) that on average the total number of defects increases with energy , but it shows strong local variations . only an even number of defects are obtained , because the net topological charge is always six , and the dipole defects ( i.e. one dislocation with ` 1 ' and ` + 1 ' defects ) always appear in pairs . also the hexagonal position of the defects disappears ( see fig .  [ fig3](1 ) ) and more free dislocations are found . these defects move from the transition region to the border ( see fig .  [ fig3](2 ) ) or to the central region ( see fig . [ fig3](3 ) ) . for configurations with higher energy , the defects arrange themselves in long chains , i.e. dislocation lines . on average the configurations with defects at the border have a lower energy than those with defects in the center . we also investigated whether or not it is possible to have a configuration with only six 5-fold disclinations and no other defects ( like for example for the @xmath39 configuration with 24 particles at the outermost ring @xcite ) . therefore , we started our mc procedure with a perfect hexagonal structure without any defect and then allowed it to relax to its energy minimum . we did this for @xmath40 up to @xmath41 particles , because we noted that for these particle numbers the configuration has about 42 particles in the outer ring , which is a multiple of six , i.e. the net topological charge . only in such a case one can have the situation in which just six 5-fold disclinations are present . we found that our result ( from @xmath40 up to @xmath41 particles ) never converges to a configuration with only six 5-fold disclinations . however , this procedure indeed favorably relaxes to configurations with 42 particles in the outer ring , often resulting in a configuration which has less total number of defects than the corresponding groundstate . the effect of the geometry induced defects on the eigenfrequencies ( i.e. the eigenmode spectrum ) were also investigated for these large clusters . in this system , it is well known that there are three eigenfrequencies which are independent of @xmath13 @xcite : @xmath42 and @xmath43 , which correspond to the rotation of the system as a whole , the center of mass mode and the breathing mode , respectively . the above modes were recently observed experimentally @xcite . the smallest frequency no longer correspond to intershell rotation as in small clusters @xcite but to the excitation of a vortex / antivortex pair , of which a typical mode is shown in fig . [ fig4](a ) . slightly larger excitation energies may consist of multiples of such pairs ( see fig . [ fig4](b ) ) . modes with higher eigenfrequencies often show a hexagonal structure similar to the ordering of the defects . the motion can be concentrated around ( see fig .  [ fig4](c ) ) or between the defects ( see fig . [ fig4](d ) ) . the local modes can be found at the six corners of the hexagon where the defects are exactly situated ( see fig .  [ fig4](e ) ) . the modes in which the inner particles have larger amplitudes than the outer particles have the largest eigenfrequencies ( see fig . [ fig4](f ) ) . the lowest eigenfrequencies of the excitation spectrum corresponding to the ground - state configuration of the system is shown in fig . [ fig5 ] , as function of the number of particles for @xmath13 ranging from @xmath44 to @xmath45 . the labels in fig . [ fig5 ] denote the total number of defects present in the ground state . notice that only an even number of defects are obtained as explained before . on average , configurations with a large number of defects have a smaller lowest eigenfrequency and are thus less stable , and vice versa .    in this 2d lattice , all behaviors of the cluster modes can be classified as shear - like or compression - like modes . in order to characterize the compressional and shear parts of these eigenmodes , we calculated respectively the divergence @xmath46 and the vorticity @xmath47 of the velocity field . to calculate the velocity field , we interpolated the displacements of fig . 4 on a @xmath48 grid ( thus neglecting the constant eigenfrequency ) . the `` divergence and vorticity maps '' were then calculated at every point of this matrix . notice that pure shear or compressional modes do not exist in the circular boundary of finite cluster . 6(b ) shows the vorticity and thus displays the shear part of the eigenmode of fig . the two vortex / antivortex pairs are clearly seen . the `` divergence map '' for this eigenmode is practically zero everywhere , as there is no compressional part ( fig . this is not the case for the eigenmodes ( c ) and ( d ) of fig . 6(c ) and ( e ) show the `` divergence maps '' for both eigenmodes , in which the compression and rarefaction can be clearly seen . both cases show no shear part ( figs . 6(d ) and ( f ) ) . figs . 6(c ) and 6(e ) ( see also the 3d plots at the bottom of fig . 6 ) exhibit clearly dipole type of compressional oscillations between ( fig . 6(c ) ) and at ( fig . 6(e ) ) the defect regions . understanding the microscopic mechanism of melting has intrigued scientists since the late nineteenth century . special interest has been devoted to 2d melting @xcite . most works address infinite systems consisting of a single layer . however , whether melting of a 2d crystal is a first order transition and proceeds discontinuously or is a continuous transition in which the crystal first transits into a hexatic phase retaining quasi - long - range orientational order and then melts into an isotropic fluid , is still an open question and a controversial issue .    in the present work we consider a finite 2d system where we take @xmath23 for our numerical simulation . here we present a calculation of the melting phase diagram by performing mc simulations . in ref . @xcite molecular dynamics was used to investigate the melting of a cluster of particles interacting through a logarithmic interaction . as compared to our coulomb interaction where the geometry induced defects are situated in the 3@xmath49 and 4@xmath49 outer shells ( i.e. the transition region ) and around the six corners of the `` defect '' hexagon , in the logarithmic interacting system @xcite those defect are mainly situated in the outer two shells . in ref . @xcite , the number and type of defects were studied as function of the noise ( i.e. temperature ) . here we will use several different criteria such as the total energy , the radial dependent mean square displacement , the bond - angular order factor and the angular square deviation to characterize the melting behavior of the cluster . there are several different criteria that can be used to find the melting temperature . in order to determine the melting transition point , we calculated the potential energy of the system as a function of temperature ( see fig . [ fig7 ] ) . in the crystalline state the potential energy of the system increases almost linearly with temperature and then after the critical temperature is reached ( @xmath50 for @xmath23 ) , it increases more steeply as shown in fig . this is a signature of melting and is related to the unbinding of dislocation pairs . the dotted assurgent line in fig . 7 indicates the linear temperature dependence of the potential energy for low temperatures before melting . in the upper inset fig . [ fig7](a ) , we plot @xmath20 which is the difference between the numerical obtained energy and the linear t - behavior . after the melting point , @xmath51 increases sup - linear . the lower inset , fig . [ fig7](b ) , shows the averaged number of defects as function of temperature @xmath52 . the number of defects were obtained as follows . we considered 40 configurations for every temperature @xmath52 . every 500 mc steps a new configuration was obtained . for all these configurations the number of defects were counted . finally we averaged over the 40 configurations , which is the reason why the number of defects can be non - integer . with increasing temperature , the system generates more and more defects and after the melting point the defect number grows very fast . notice that two clear critical temperatures emerge from this figure at the crossing points of the dotted lines , i.e. @xmath50 and @xmath53 .    in fig . 8 we plot typical particle trajectories for different temperatures which shows that the melting of this system is very complex and non - homogeneous . it clearly indicates that the melting starts around the six corners of the hexagon which are exactly the defect regions . with increasing temperature , the particles in the defect region start to move radially and destroy order locally . with further increase of temperature the total system completely melts and the order is destroyed .    in order to better describe the spatial dependence of the melting process in our system , we separate the configuration into three regions as shown in fig .  [ fig2 ] . region i ( dark grey colored hexagonal area ) is comprised of the defect - free hexagonal center ; region ii is a transition region with the defects ( light grey colored area ) , and region iii consists of the outermost two rings . for the case of @xmath23 particles region i consists of @xmath54 particles , region ii  consists of @xmath55 particles and region iii  of @xmath56 particles . we calculate for each region the mean square displacement @xmath57 , which was introduced in ref . @xcite ,    @xmath58    with @xmath59 the average distance between the particles . [ fig9 ] shows the @xmath60 as a function of the reduced temperature @xmath52 for the three different regions . at low temperatures the particles exhibit harmonic oscillations around their @xmath61 equilibrium position , and the oscillation amplitude increases linearly and slowly with temperature : the particles are well localized and display still an ordered structure . this linear dependence is accentuated by the thin straight lines in fig . , we already notice that the amplitude of the local particle thermal vibrations in these different regions are different . the amplitude is largest at the defect region and lowest in the center of the cluster . melting occurs when @xmath60 increases very sharply with @xmath15 . because of the finite number of particles one has rather a melting region , instead of a well - defined melting temperature . after the melting `` point '' , the particles exhibit liquid - like behavior . [ fig9 ] exhibits three different melting temperatures corresponding to the three different regions . firstly region ii , i.e. the transition region containing the defects , starts to melt , then the outermost two rings melt , and finally the hexagonal region melts . following ref . @xcite , we can `` define '' a melting temperature at the point where @xmath62 which results into the melting temperatures @xmath63 and @xmath64 for the defect region , the outer rings and the center region , respectively .    in order to investigate the melting in the defect region in further detail we consider two new small regions as shown in the inset of fig . [ fig10 ] . one region is around the defect , the other does nt contain a defect and is situated between two defect regions . for @xmath23 each of the two regions contains respectively eight and seven  particles . in fig . [ fig10 ] , the @xmath57 of these two different regions show a different melting temperature : the melting clearly starts first around the defect as expected . the particle motion is strongly influenced by the topological defects , i.e. the particles in the defect regions are less well interlocked and have a larger diffusion constant than the undistorted lattice regions and their thermal motions are easier to be excited @xcite . notice , that for the two separate regions a much sharper melting behavior is found than for the intermediate region as a whole ( see fig . [ fig9 ] ) . the reason of course is that in fig . [ fig9 ] one averages over defect and defect - free regions . the criterion @xmath65 results into @xmath66 for the defect and the defect - free regions , respectively . these two melting temperatures are very similar to the melting temperature of the transition region and the hexagonal region of fig . the third independent parameter is the bond - orientational correlation function . this quantity determines the type of melting transition and the melting point for an infinite system . our finite system is too small in order to have a reliable analysis of the asymptotic decay of the density correlation function . therefore , we calculate the bond - angular order factor which was originally presented in ref . @xcite , but following ref . @xcite we modified it into ,    @xmath67    this quantity is calculated only for region i which exhibits a hexagon structure , where @xmath68 means the @xmath69 nearest neighbors of particle @xmath70 , for ideal hexagonal lattice @xmath71 , where @xmath72 is the angle between some fixed axis and the vector which connects the @xmath73 particle and its nearest @xmath74 neighbor .    for a perfect hexagonal system @xmath75 in our system for @xmath23 , the initial value of @xmath76 is 0.96 , which means that the structure in region i  is almost perfect hexagonal . our numerical results ( see open dots in fig . [ fig9 ] ) show that @xmath77 decreases linearly with increasing temperature . when @xmath76 is around @xmath78 , it more rapidly drops to zero with increasing temperature . @xmath76 should be zero for the liquid state . this can be compared with the infinite system where a universal melting criterion was found in ref . @xcite : melting occurs when the bond - angle correlation factor becomes @xmath79 @xmath80 , which was found to be independent of the functional form of the interparticle interaction . for our system the value @xmath79 @xmath80 is probably not correct because in our finite system @xmath76 does not drop to zero at @xmath81 , but is smeared out around @xmath81 . therefore , the midpoint @xmath82 is expected to describe better the melting temperature . this leads to @xmath83 which is similar to the result @xmath84 obtained from the radial displacement criterion . in contrast to bulk systems , the melting scenario of small laterally confined 2d systems was found earlier @xcite to be a two step process . upon increasing the temperature , first intershell rotation becomes possible where orientational order between adjacent shells is lost while retaining their internal order and the shell structure . at even higher temperatures , the growth of thermal fluctuations leads to radial diffusion between the shells , which finally destroys the positional order . to characterize the relative angular intrashell and the relative angular intershell , we use the functions as defined in ref . the relative angular intrashell square deviation    @xmath85 /\varphi _ { 0}^{2},\ ] ]    and the relative angular intershell square deviation    @xmath86 /\varphi _ { 0}^{2},\ ] ]    where @xmath87 indicates the nearest particle from the same shell , while @xmath88 refers to the nearest - neighbor shell , @xmath89 where the number in the outermost two rings @xmath90is the same and equals 42 for our @xmath23 system . only the two outermost rings have a clear shell structure . both two outer rings are strongly interlocked which is a consequence of the 1d wigner lattice arrangement of the two rings . from the inset of fig . [ fig11 ] , one can see that the inner ring will melt before the outermost ring . we find that the result for @xmath91 of the inner ring is almost the same as @xmath92 which is the relative angular intershell square deviation . it means that when the inner ring loses its order , the relative order is lost simultaneously . the outermost ring can still keep its order and it will melt at even higher temperature . comparing this with fig . 9 , we see that the radial and angular displacements start to increase rapidly at approximately the same temperature . thus for large clusters intershell rotation will not occur below the melting temperature , but appears at the same temperature when the radial displacements increase . the configurational and melting properties of large two - dimensional clusters of charged classical particles interacting with each other via the coulomb potential were investigated through the monte carlo simulation technique . for the ground state configuration , a hexagonal wigner lattice is formed in the central area while at the border of the cluster the particles are arranged in rings . in the transition region between them defects appear as groups of dislocations and disclinations at the six corners of the hexagonal - shaped inner domain . many different arrangements and types of defects are possible as metastable configurations with a slightly higher energy . the particles motion is found to be strongly related to the local topological structure . our results clearly show that the melting of the clusters starts near the geometry induced defects , and that three melting temperatures can be obtained : @xmath63 and @xmath93 for the defect region , the outer rings and the center region , respectively . these values are for the @xmath23 cluster . taking a different value for @xmath13 does not lead to any qualitative differences , it only influences slightly the values for the three melting temperatures . this work was supported by the flemish science foundation ( fwo - vl ) , the belgian inter - university attraction poles ( iuap - v ) , the `` onderzoeksraad van de universiteit antwerpen''(goa ) , and the eu research training network on `` surface electrons on mesoscopic structures '' . we are very grateful to dr . i. v. schweigert for helpful discussions . stimulating discussions with prof . a. matulis and m. miloevi are gratefully acknowledged . b. partoens is a post - doctoral researcher of the flemish science foundation ( fwo - vl ) .
the configurational and melting properties of large two - dimensional ( 2d ) clusters of charged classical particles interacting with each other via the coulomb potential are investigated through the monte carlo simulation technique . 
 the particles are confined by a harmonic potential . for a large number of particles in the cluster ( n150 ) 
 the configuration is determined by two competing effects , namely in the center a hexagonal lattice is formed , which is the groundstate for an infinite 2d system , and the confinement which imposes its circular symmetry on the outer edge . as a result 
 a hexagonal wigner lattice is formed in the central area while at the border of the cluster the particles are arranged in rings . in the transition region defects 
 appear as dislocations and disclinations at the six corners of the hexagonal - shaped inner domain . 
 many different arrangements and type of defects are possible as metastable configurations with a slightly higher energy . the particles motion is found to be strongly related to the topological structure . 
 our results clearly show that the melting of the clusters starts near the geometry induced defects , and that three different melting temperatures can be defined corresponding to the melting of different regions in the cluster . 
 pacs numbers : 45.05.+x , 61.46.+w , 73.22.-f