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melting of crystals has always been a fascinating subject for more than a century since the discovery of the empirical lindemann s criterion @xcite .
the lindemann s criterion says that if the average of vibration amplitude u , namely @xmath0 , exceeds a certain value , usually 10@xmath1 of the distance between nearest - neighbors , then the melting occurs . times and over again , many authors have tried to find out microscopic precursor mechanisms that lead a crystal to melt . until 30 years ago , one of the favorite pictures of melting is the softening of a phonon mode due to the temperature ( t ) .
the atoms have no longer restoring forces which keep them staying close to their equilibrium positions : they move around and the system goes to a liquid state .
the soft - mode picture has encountered some scepticism because in real crystals as well as in simulations one observes that well below the melting temperature ( @xmath2 ) , many defects , dislocations , interstitial atoms ... are excited .
therefore , it is hard to believe that the system stays in a periodic structure with propagating phonon modes up to @xmath2 .
evidence of defects is found in many works @xcite .
another question that is unsolved in a clear manner is the form of the potential that binds the atoms together in a given lattice structure . in a microscopic point of view
, the potential should come mainly from the symmetry of atomic orbitals .
but ab - initio calculations are still far away from being able to use realistic hypotheses @xcite .
empirical potentials have been used instead to study melting .
one can mention the popular 6 - 12 power lennard - jones ( lj ) potential @xcite , various similar power potentials , the many - body gupta s potential @xcite , the stillinger - weber ( sw ) potential @xcite , and the tersoff potential @xcite .
two - body potentials such as the lj one crystallize atoms in the fcc at low temperatures and nothing else ; this comes from the fact that lj potential is isotropic so the atoms are crystallized in the most dense isotropic structure , namely the fcc lattice . in order to stabilize other structures , several phenomenological potentials have been introduced , often without a microscopic justification .
for example , the sw potential or the tersoff potential stabilize the diamond structure at low temperatures .
these potentials have been used with success to calculate properties of si clusters @xcite and amorphous si crystals @xcite .    in this paper
, we use the lj potential to study the melting of rare - gas crystals which have the fcc lattice structure at low temperatures .
it is amazing that such a simple question was not studied with precision so far in spite of an abundance of experimental data on rare gas such as ar , xe , ne and kr .
most of the melting studies concerning rare gas were done in particular cases : small clusters @xcite , adlayers on a substrate , etc . the main reason to avoid to study
the bulk melting may be due to some technical difficulties such as periodic boundary conditions , volume expansion with temperature , etc .
previous monte carlo ( mc ) studies of bulk melting have been carried out with lj potential but emphasize was put on the melting mechanism rather than on the precise melting temperature in real materials @xcite .
the purpose of this paper is therefore to test whether or not the experimental @xmath2 can be reproduced by mc simulation using the values of the lj parameters deduced for rare gas in the gaseous state long time ago@xcite .
we will show here that by appropriate choices of technical procedures , we are able to obtain melting temperature for various rare gas _ directly _ from our simulations , unlike previous simulations @xcite which have had recourse to various means and some thermodynamic functions to deduce it .
we find in this work the melting temperatures for several rare gas higher than experimental values .
a revision of the values of lj parameters widely used in the literature for more than 50 years should be made in order to better describe the solid state of rare gas .
note that in a recent work @xcite , a hypothetical thermodynamic integration path is used to find the relative free energies of the solid and liquid phases , for various system sizes , at constant cutoff radius , in an attempt to explain the overestimate of the melting temperature with the lj potential .
however , due to various approximations , several results were not physically clear , in particular why the melting temperature oscillates with increasing cutoff distance .
section [ sec : level2 ] is devoted to a description of the model and our mc technique .
the results are shown and discussed in sect .
[ sec : level3 ] .
concluding remarks are given in sect .
[ sec : level4 ] .
the lj potential is given by @xmath3 with @xmath4\ ] ] where @xmath5 , @xmath6 , @xmath7 and @xmath8 are constants , @xmath9 denotes the distance between two atoms at @xmath10 and @xmath11 . here
@xmath12 and @xmath13 .
we list for convenience the values of the above constants for various rare gas and their melting temperatures , using the data from ref .
note that the constants @xmath5 and @xmath8 are those deduced with some approximations using experimental data of the gaseous phase of rare gas which have been described in the pioneer paper of bernardes@xcite .
the listed values of the constants should be therefore viewed as approximate values . to our knowledge
, there were not papers using these constants to verify if the melting temperature is correctly obtained , either by theoretical calculations or by simulations .
the absence of works dealing with this point has motivated our present work .    in one of our previous works @xcite , we have used the lj potential to investigate the mechanism which initiates the melting .
we have counted the number of defects created with increasing temperature ( @xmath14 ) and found that the melting occurs when these defects interact with each other forming a kind of chains of defects which break the solid periodic state at a temperature well below the melting point . in that work
, we did not use the constants of the lj potential for any specific material .
we used instead the reduced temperature and dimensionless parameters .
it was not our purpose to clarify the value of melting temperature for a given kind of crystal .
it is now the time to verify those constants to see whether or not they yield the correct value of @xmath2 for the melting of the solid phase of rare gas .
( color online ) the lennard - jones potential for ar with the parameters listed in table [ tab : table1].,width=302,height=188 ]    .[tab : table1]lenard - jones parameters [ cols= " < , < , < , < " , ]     at this stage , it is interesting to note that molecular dynamics ( md ) simulation of melting of a perfect crystal with periodic boundary conditions produce superheating .
there is an empirical rule which states that the melting temperature of a crystal without any surface and any defects , is @xmath15higher than the true thermodynamic melting temperature @xmath2 .
however in mc simulations , defects and dislocations are naturally created in the crystal by means of random numbers used in every mc step for atom displacements .
thus , the superheating should not exist .
have shown in ref .
@xcite that for ar , with mc simulation , @xmath2 is about @xmath16 higher than the experimental value . with our results for ar , we find an increase of about @xmath17 . for kr
, we find an increase of @xmath18 for @xmath19 atoms after @xmath20 mc steps per atom .
this increase is more important if we consider larger sizes as seen in the case of ar . as said ,
the high values of @xmath2 in mc simulations are not due to the superheating as in md simulations .
rather , we believe that these high values are due to the inaccuracy of the listed bernardes parameters . we will propose a modification in the following      in order to reduce the @xmath2 value , we propose now to modify the value of @xmath5 , the prefactor of the lj potential , and the coefficient @xmath8 .
note that in papers dealing with melting in other materials by means of md calculations or mc simulations , there have been several propositions to modify constants appearing in potentials in order to obtain a correct agreement with experimental value of @xmath2 .
this is because these constants are often deduced from experimental data which are not valid for the whole temperature range . among these papers
, we can mention the case of melting of si crystal studied by md @xcite and mc simulations @xcite , using the tersoff potential @xcite .
our proposition to modify some constants of the lj potential when applied to a rare - gas crystal is certainly a necessity in order to reproduce the experimental @xmath2 .
we have done simulations with different pairs of @xmath21 ) .
it turned out that @xmath2 depends essentially on @xmath5 .
there is however an optimal value of @xmath8 which is @xmath22   corresponding to the experimental nearest - neighbor distance @xmath23   of ar ( cf .
table [ tab : table5 ] ) .
we show in fig .
[ epsilon_effect ] the curves obtained for two selected values of @xmath240.008767853 and 0.008951794 which give respectively @xmath25 k and 86 k. these values of @xmath2 are in agreement with the experimental value 84 k within statistical errors .
note that the modified @xmath5 is about 15% smaller than the original bernardes value @xmath26 .
( color online ) energy per atom versus @xmath14 for two selected values of @xmath5 , with @xmath22  , for an ar crystal with @xmath27 .
the arrows indicate @xmath2 for the two indicated values of @xmath5 .
see text for comments .
, width=302,height=188 ]      at this stage , a natural question we ask ourselves is `` what is the effect of the cutoff distance @xmath28 ? '' .
we know that for a long - range interaction , the longer the interaction range is the lower the energy becomes . as a consequence ,
the melting transition is higher .
however , as @xmath28 increases , the contribution of neglected neighbors becomes smaller . from a certain value of @xmath28 ,
@xmath2 does not vary significantly .
this is observed in fig .
[ rc_effect ] where @xmath2 is saturated for @xmath29 , i. e. @xmath30  .
all the results shown above for @xmath28 are valid in the discussion of the size effect and the modification of @xmath5 and @xmath8 .
( color online ) energy per atom versus @xmath14 with various values of @xmath28 for ar crystal with @xmath27 .
the left arrow indicates @xmath2 for @xmath31 and the right arrow indicates @xmath2 for @xmath32 and @xmath33  .
note that @xmath34 is the fcc lattice constant which is equal to @xmath35 where @xmath23   is the nn distance .
see text for comments .
, width=302,height=188 ]      in order to show that our algorithm works well with other rare gas , we have plotted the curve of energy versus temperature , obtained for krypton and xenon in fig .
[ evtkr ] .
again here , we see that @xmath2 , even for a small size , is already higher than the experimental value for each crystal .
we think that the bernardes parameters for kr and xe should be modified to get an agreement with experiments as what proposed above for ar .
( color online ) upper : energy versus temperature for a krypton perfect crystal with @xmath19 atoms .
one observes that @xmath36 k while the experimental value is @xmath37 k. lower : energy versus temperature for a xenon perfect crystal with @xmath27 atoms .
one observes that @xmath38 k while the experimental value is @xmath39 k. these curves have been obtained with the bernardes values of parameters . ,
title="fig:",width=302,height=188 ] ( color online ) upper : energy versus temperature for a krypton perfect crystal with @xmath19 atoms .
one observes that @xmath36 k while the experimental value is @xmath37 k. lower : energy versus temperature for a xenon perfect crystal with @xmath27 atoms .
one observes that @xmath38 k while the experimental value is @xmath39 k. these curves have been obtained with the bernardes values of parameters .
, title="fig:",width=302,height=188 ]
we have shown in this paper results of the melting temperature for rare gas , by performing extensive mc simulations with the lj potential .
we have obtained directly from our simulations physical quantities such as internal energy , lattice constant and radial distribution as functions of temperature .
we have shown that melting occurs with a large latent heat and a jump in the lattice constant .
effects of system size and cutoff distance have been investigated .
let us emphasize that the theoretical lj parameters widely used in the literature @xcite yield a melting temperature higher than the experimental one as seen above , from @xmath40 15% for kr to @xmath40 20% for ar .
this is not a surprise because those lj parameters already yield theoretical nn distance , cohesive energy and bulk modulus different from corresponding experimental ones ( see table [ tab : table5 ] ) .
we have demonstrated that , in order to reduce the melting temperature to fit with experiments , it is necessary to modify the original bernardes lj parameter @xmath5 in such a way to reduce the energy at @xmath41 .
the effect of @xmath8 , within possible values of nn distance , is very small on @xmath2 .
a good agreement on @xmath2 between experiments and simulations for ar is obtained with the modified values given in fig .
[ epsilon_effect ] . for other rare gas such as kr and xe
, we have to proceed to a modification of their barnardes parameters in a manner similar to that done above for ar if we want to get a good agreement with experiments .
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we study the melting transition in crystals of rare gas ar , xe , and kr by the use of extensive monte carlo simulations with the lennard - jones potential . 
 the parameters of this potential have been deduced by bernardes in 1958 from experiments of rare gas in the gaseous phase . 
 it is amazing that the parameters of such a popular potential were not fully tested so far . in order to carry out precise tests , 
 we have written a high - performance monte carlo program which allows in particular to take into account correctly the periodic boundary conditions to reduce surface effects and to reduce cpu time . using the bernardes parameters , we find that the melting temperature of several rare gas is from 13 to 20% higher than that obtained from experiments . 
 we have throughout studied the case of ar by examining both finite - size and cutoff - distance effects . in order to get a good agreement with the experimental melting temperature , 
 we propose a modification of these parameters to describe better the melting of rare - gas crystals .    
 pacs numbers : 34.20.cf , 64.70.dv , 65.20.+w , 65.40.-b : :