VOLUME 44, NUMBER 7 PHYSICAL REVIEW LETTERS 18 FEBRUARY 1980

spectrum.In conclusion we have established the existence

of twelve orthorhombic domains in NaCN. Wehave determined their relative orientation bothto the pseudocubic phase and to each other.

7;-'e wish to thank M. DeLong of the Universityof Utah Crystal Growth Laboratory for supplyingthe crystals and Dr. C. E. Hayes for assistancewith the apparatus. Many useful discussions oc-curred with Dr. H. T. Stokes, Dr. L. C. Scavardado Carmo, Dr. P. Gash, and especially Profes-sor F. Luty. We are particularly indebted toTom Case for aiding us with the computer simu-lation of the data.

On sabbatical leave from the Bacah Institute ofPhysics, The Hebrew University of Jerusalem, Jeru-salem, Israel.

'J. A. Lely, dissertation, University of Utrecht, 1942( p bli hed), and unp bli h d.

2H. J. Verweel and J. M. Bijvoet, Z. Kristallogr.Kristallgeom. Kristallphys. Kristallchem. 100, 201

(1940).3M. J. Bijvoet and J. A. Lely, Rec. Trav. Chim.

Pays-Bas 59, 908 (1940).4A. Cimino, G. S. Parry, and A. R. Ubbelohd, Proc.

Roy. Soc. London, Ser. A 252, 445 (1959).5A. Cimino and G. S. Parry, Nuovo Cimento 19, 971

I'1961) .6G. S. Parry, Acta Crystallogr. 15, 596 (1962).G. S. Parry, Acta Crystallogr. 15, 601 (1962).

~Y. Kondo, D. Schoemaker, and F. Liity, Phys. Rev.B 19, 4210 (1979).

D. Fontaine, C. R. Acad. Sci. Paris, Ser. B 281,443 (1975).' J. M. Rowe, J. J. Rush, and E. Prince, J. Chem.

Phys 66 5147 (1977)"D. E. O'B,eilly, E. M. Peterson, C. E. Schlie, and

P. K. Kadaba, J. Chem. Phys. 58, 3018 (1973).J. F. Hon and P. J. Bray, Phys. Rev. 110, 624

(1958)."T.P. Das and E. L. Hahn, in Nuclear Quadrupole

Resonance Spectroscopy, Suppl. No. 1 to Solid StatePhysics, edited by H. Ehrenreich, F. Seitz, andD. Turnbull (Academic, New York, 1958).'4G. H. Stauss, J. Chem. Phys. 40, 1988 (1964).

Melting in Two Dimensions is First Order: An Isothermal-Isobaric Monte Carlo Study

Farid F. AbrahamIBM Research Laboyatot'y, San Jose, California 95I99

(Received 19 November 1979)

Isothermal-isobaric Monte Carlo computer experiments on melting in a two-dimen-sional Lennard-Jones system indicate that the transition is first order, in contrast tothe two-stage, second-order melting behavior suggested as a possibility by Halperinand Nelson.

Expanding on the proposals by Kosterlitz andThouless' and by Feynman, '" Halperin and Nel-son4" have developed a detailed theory of dislo-cation-mediated melting for a two-dimensional"crystal. " One important feature of the Halperin-Nelson theory is the possibility that the transi-tion from two-dimensional solid to two-dimen-sional liquid takes place by tzvo second-olde~transitions with increasing temperature. At sometemperature T, dissociation of dislocation pairsgives rise to a second-order transition from asolid phase, with algebraic decay of translationalorder and long-range orientational order, to a"litluid-crystal" ("hexatic") phase, with expo-nential decay of translational order but algebraicdecay of sixfold orientational order. At a highertemperature T~ », dissociation of dislocationsinto disclinations gives rise to another second-order phase transition from the hexatic phase to

the isotropic Quid phase. Halperin and Nelsondo emphasize that this particular melting mech-anism is only one possibility. They cannot ruleout the possibility of a first-order melting transi-tion.

Direct experimental verification of the Halper-in-Nelson theory for two-dimensional meltingis difficult because several possible mechanismsare involved in real systems which might conceiv-ably influence the apparent order of the transi-tion; e.g. , epitaxy, second-layer promotion, andheterogeneity, as well as the details of the adat-om and substrate interactions. ' In order to cir-cumvent the uncertainties and limitations of cur-rent laboratory experiments, Frenkel and Mc-Tague' performed a "computer experiment" onthe well-defined model system of Lennard-Jones(L-Z) 12:6 atoms constrained to remain two di-mensional. They chose the molecular-dynamics

1980 The American Physical Society 463

VOLUME 44, NUMBER 7 PHYSICAL REVIEW LETTERS 18 FEBRUARY 1980

simulation method (with imposed periodic bound-ary conditions to simulate an infinite plane) tostudy the structure and thermodynamics of 256L-J atoms at a reduced density p* =0.80 and re-duced temperature range of 0.25 ~ T*&1.25.From an analysis of their experimental data,Frenkel and Mc Tague concluded that the L-Jsystem loses its resistance to shear at a tern-perature T *=—0.36, but has long-range "orienta-tional" order up to a higher temperature T;~0.57; hence their molecular-dynamics experi-ments support the two-dimensional melting the-ory of Halperin and Nelson.

Attracted by the unusual predictions of the the-ory and the apparent confirmation by computerexperiment, I decided to look more closely atthe therrnodynanzics of the two-dimensional melt-ing phenomenon by doing a different type of com-puter experiment: By employing the isothermal-isobaric Monte Carlo simulation method of class-ical statistical mechanics, '"we may calculatethe equilibrium energy, entha/py, and density ofthe existing phase (whether solid, liquid, or hex-atic) for a chosen temperature and pressureHence, by performing a series of experimentsat fixed pressure and for a range of tempera-tures, we should observe (a) discontinuities inenthalpy and density at T if the melting transi-tion is first-order, or (b) discontinuities in thetemperature derivatives of the enthalpy and den-sity at T and T; if the melting transitions aresecond order. Also, as pointed out by Frenkeland Mc Tague, there should (should not) be hys-teresis if the system passes back through the ap-parent melting temperature when the phase tran-sition is first order (second order). I have per-formed such experiments and conclude that two-dimensional melting is a first-order phase tran-sition.

The (N, P, T) Monte Carlo calculations wereperformed on an N = 256 atom system with peri-odic boundary conditions to simulate the bulk.The interatomic force law was taken to be L-J12:6 with (e, a) denoting the well-depth and sizeparemeters, respectively. In a typical simula-tion "experiment, " the system was "equilibrated"through 10' Monte Carlo moves with a 50% ac-ceptance ratio. After equilibration, 10' to 2x10'further moves were performed to obtain the en-thalpy H, the average density p, the pair dis-tribution function g(r), and other quantities.Great care was practiced to determine that thesystem was in 'local equilibrium" (stable ormetastable) when taking the statistics for the

e0.9 -~

oeI

0.8 -~~ ~

CV

Entha I py

c(D I

0.6—

— 3.50

- 3.25

- 3.00

2.75

250 c

— 2.25

0.5—— 2.00

— 1.75

0.4 I I I I

02 03 04 05 06Temperature, kT/e

FIG. 1. The equilibrium density and enthalpy peratom as a function of temperature for the I,ennard-Jones system at a fixed pressure Pa/e =0.05..

quantities of interest.In Fig. 1, I present the equilibrium density

p* =pa' and the enthalpy per atom h* =H/Ne forthe L-J system as a function of temperatureT* =hT'/e and for a fixed pressure P*=Pa'/e= 0.05. The pressure was intentionally chosento be low since Nelson and Halperin have specu-lated that the melting transition could be firstorder at high pressures but second order at lowpressures, 4 and since it was suggested4" thatearlier studies' "were examining a high-(tem-perature, density) regime where melting be-comes first order. " Considering first the den-sity behavior, note that the solid density decreas-es smoothly as I sequentially increase the tem-perature T* =0.46. At T* =0.48, the solid meltsinto a liquid with a dramatic decrease in equilib-rium density p, * -p&* =0.11. The pair distribu-tion functions and "snap-shot" pictures of theatomic configurations (see Fig. 2) show the changefrom crystalline order to liquid disorder. Athigher temperatures, the liquid density decreas-es smoothly. By sequential decrease of temper-ature and equilibration, the system passesthrough T* = 0.48 and remains a liquid withsmoothly increasing density down to a temper-ature T* =0.43 (again, the pair distribution func-tions reflect the fact that the undercooled statesare liquid). This establishes hysteresis whenpassing back through the apparent melting tem-per~re. At T* =0.4, the liquid solidifies with

464

VOLUME 44, +UMBER 7 PHYSICAL REVIEW LETTERS 18 I'EBRUWRV 1980

T= 0.4

Solid

Ojg4( ~~

0.45 --P y 0.46

~ eA~oCeA oo

I Melting ',

plQ. 2. Snapshot pictures of the 256-atom L-J system simulated by the isothermal-isobaric Monte Carlo methodfor the denoted temperatures and a fixed pressure of Po /e =0.05.

a sharp increase in density p,* =0.8; however,this density is lower than the original solid atthat temperature, p,* =0.85. Examination of theatomic structure of this solid, which was nucleat-ed from the liquid phase, revealed significant de-fect structure which accounts for the density dif-ference for the two solid systems. Of course, itwould be unlikely that the supercooled liquidwould solidify directly into a defect-free crys-tal structure. The subsequent time for relaxa-tion to a defect-free solid-state structure is toolong to follow by computer simulation, and, inany case, this relaxation process is irrelevantto the conclusions of this study. The enthalpyof the system has the same behavior as the den-sity and yields a latent heat of ™0.44m/atom.

The density discontinuity between solid and li-quid, the evident latent heat, and the existenceof metastability certainly demonstrate that melt-ing of a two-dimensional solid is a first-orderphase transition.

I conclude with the following comment. Oneshould not be too concerned with the statementthat the existence of the crystalline solid statein two dimensions is impossible. "'~ I explainmy viewpoint. At a nonzero temperature, theperfect translational order of the atoms in aclassical crystal is destroyed because of therandom thermal motion of the atoms about theirequilibrium positions. Furthermore, in contrastto a 3-D (three-dimensional) crystal, the atomicequilibrium positions themselves in a 2-D crys-tal become uncorrelated at large separations. ""A quantitative measure of the loss of long-rangecrystalline order is the difference 6 betweenthe average separation between two atoms and

the distance corresponding to the proper numberof lattice spacings. It is found that & divergesslowly with N, the number of atoms in the 2-Dsystem, the dependence on N being only logarith-mic. Using a relation found by Hoover, Ashurst,and Olness, "I estimate that to lose crystallinecorrelation equal to one lattice spacing (6- 3 A)near the melting point, the area of the 2-D crys-tal should be - 10 ' cm', very large indeed. Atthe uppermost extreme, a crystal of dimensionequal to the size of the universe (- 10"light-years) would have a correlation loss of -6 A!"I can only conclude that for real situations, thecrystalline solid state in two dimensions canexist. In reality, it will be the imperfectness ofthe material state (e.g. , grain boundaries), andnot a logarithmic divergence, that will preventperfect translational order over macroscopicdimensions. The fact that "for practical pur-poses" the 2-D crystalline state exists suggeststhat Landau's argument that "transitions betweenbodies of different symmetry (in particular be-tween a liquid and a crystal) cannot happen con-tinuously"" may well be applicable in two di-mensions.

I thank J. A. Barker, D. Henderson, J. Hub-

bard, and M. Warner for discussions and forcritically reading the manuscript.

J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6,1181 (1973).

R. P. Feynman, unpublishai. An outline of thistheory is given by R. L. Elgin and D. L. Goodstein,Phys. Rev. A 9, 2657 (1974), and by J. G. Dash, Phys.Rep. 38C, 177 (1978).

465

VOLUME 44, NUMBER 7 PHYSICAL REVIEW LETTERS I8 FEBRUARY 1980

B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41,121, 519(E) (1978).

4D. R. Nelson and B. I. Halperin, Phys. Rev. B 19,2457 (1979).

5J. G. Dash, Phys. Rep. 38C, 177 (1978).D. Frenkel and J. P. McTague, Phys. Rev. Lett. 42,

1632 (1979).~W. W. Wood, in Physics of Simple Fluids, edited by

H. ¹ V. Temperley, G. S. Rushbrooke, and J. S. Row-linson (North-Holland, Amsterdam, 1968), Chap. 5.

81. R. McDonald, Mol. Phys. 23, 41 (1972).B.J. Alder and T. E. Wainrvright, Phys. Rev. 127,

359 (1962).oF. Tsien and J. P. Valleau, Mol. Phys; 27, 177

(1974) .

"S.Toxvmrd, Mol. Phys. 29, 373 (1975).~2The critical constants for the 2-D L-J system have

been estimated by Tsien and Valleau (Ref. 10) to beP~ *~0.11 and T, *& 0.625. The triple point is moredifficult to ascertain, but a "best estimate" by this au-thor indicates that the triple-point constants are P& *= 0.01 and T, *~ 0.4.

R. E. Peierls, Ann. Inst. Henri Poincare 5, 1771(1935).'4L. D. Landau, in Collected Papers of J. D. Landau,

edited by D. ter Hasr (Pergsmon, Oxford, 1965), Chap.29.

5See Dash, Ref. 5, pp. 179-183.~6W. G. Hoover, W. T. Ashurst, and R. J. Qlness, J.

Chem. Phys. 60, 4043 (1974).

Completion of the Phase Diagram for the Monolayer Regime ofthe Krypton-Graphite Adsorption System

D. M. Butler, ~'~ J. A. I itzinger, and G. A. StewartDepartment of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania 15250

(Received 4 December 1979)

The results of a heat-capacity study of the region between one and two layers in theKr/graphite (Grafoil) system are presented. Heat-capacity anomalies correspondingto the commensurable-incommensurable transition, the commensurable-Quid transi-tion, and a previously unobserved incommensurable-Quid transition delimit the high-coverage extent of the registered phase. A phase diagram proposed on the basis ofthe heat-capacity data suggests the possibility of a new type of multicritical point.

Krypton physisorbed onto graphite has beenthe subject of a number of recent studies. Iso-therm, low-energy electron diffraction, x-ray,and heat-capacity studies' have been used to mapphase boundaries in the submonolayer and mono-layer regimes. In particular, for coverages nearone registered monolayer two higher-order tran-sitions have been observed, a commensurable-incommensurable transition (CIT) at lower tem-peratures and a commensurable (registered) sol-id to a dense fluid transition (CFT). The phasediagram for the monolayer regime of the Kr/graphite system remains incomplete, however,since no transition delimiting the high-coverageextent of the registered solid has been observed.It has been conjectured' that the CIT and CFTjoin, enclosing the registered phase in a line ofhigher-order transitions and raising the interest-ing question of how the incommensurable solidmelts. We have conducted a comprehensive heat-capacity survey of the regime between one andtwo layers. The results reported here map theboundaries of the registered phase and show thatthe CIT and CFT intersect at approximately 127K and 1.5 layers. From this intersection a third

line of heat-capacity anomalies emerges with in-creasing coverage. These anomalies constitutethe first observation for the Kr/graphite systemof the incommensurable-solid-dense-fluid transi-tion (IFT), which appears to be higher order.The intersection of the CIT, CFT, and IFT sug-gests the possibility of a new type of multicrticalpoint.

In order to establish a detailed map of the phaseboundaries, sixteen coverages (i.e., fixed totalamount in the calorimeter) between 1.0 and 2.4registered monolayers were studied. The re-sults are summarized in Fig. 1, which shows theloci of heat-capacity anomalies in the tempera-ture-coverage plane. For all coverages abovethe monolayer, desorption is significant and theamount adsorbed becomes a function of tempera-ture. The heating paths for several representa-tive coverages are also shown in Fig. 1. Forcoverages just above the monolayer, the heatcapacity displays two anomalies [Figs. 2(a) and

2(b)]. With increasing coverage the lower temper-ature anomaly, at first (A» Fig. I) broad andbarely discernible, becomes larger and sharper,moving rapidly to higher temperature (AP,C,).

466 1980 The American Physical Society