**HAL Id: jpa-00211117**

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### Submitted on 1 Jan 1989

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**Free energy of low temperature antiphase structures in a** **FCC nearest-neighbour antiferromagnetic Ising model**

**by the Monte Carlo method**

### F. Livet

**To cite this version:**

### F. Livet. Free energy of low temperature antiphase structures in a FCC nearest-neighbour antiferro- magnetic Ising model by the Monte Carlo method. Journal de Physique, 1989, 50 (19), pp.2983-2990.

### �10.1051/jphys:0198900500190298300�. �jpa-00211117�

### Free _{energy } of low temperature antiphase structures in a FCC

## nearest-neighbour antiferromagnetic Ising ^{model } by ^{the Monte}

### Carlo method

### F. Livet

### Laboratoire de Thermodynamique ^{et } Physico-Chimie Métallurgique, ENSEEG, ^{B.P. } 75,

### 38042 Saint-Martin-d’Hères, ^{France}

### (Reçu ^{le 10 } février 1989, révisé le 3 mai 1989, accepté ^{le 6 } juin 1989)

### Résumé.

^{- }

### Les alliages CFC à mise

en### ordre (25 ^{et } ^{50 } %) ^{sont } modélisés par

^{un }

### hamiltonien

### d’Ising à interaction de premier ^{voisin } antiferromagnétique. Par la méthode de Monte Carlo,

^{on}

### montre que la dégénérescence ^{à } ^{T }

^{= }

^{0 des } structures à antiphases construites à partir ^{de} L12 ^{et } L10 ^{est } ^{levée } ^{à } T ~ 0. Les états les plus ^{stables } ^{sont }

^{ceux sans }

### antiphase. L’énergie ^{libre} superficielle ^{des } antiphases ^{est } ^{calculée }

^{en }

### fonction de la température ^{et } de la distance entre

### antiphases. ^{A } partir des résultats obtenus,

^{on }

### discute les diverses manières d’utiliser la méthode de Monte Carlo.

### Abstract.

^{2014 }

### Binary ordering ^{FCC } alloys of 25 and 50 at % concentrations

are ### modelled by

^{an}

### antiferromagnetic ^{nearest } neighbour Ising ^{model. } By the Monte Carlo method, it is shown that the antiphase degeneracy ^{of } L12 ^{and } L10 structures at T

^{= }

### 0 is raised at T ~ 0. The most stable states have

no### antiphases. The surface free energy of the antiphases is calculated and its variations

versus

### T and

versus### the distance between antiphases

^{are }

### studied. From the results obtained,

^{a}

### discussion is carried out of the various

uses### of the Monte Carlo method.

### Classification

### Physics ^{Abstracts}

### 64.60c

^{- }

### 05.50

^{- }

### 75.1OH

### Introduction.

### A lot of alloys ^{with } high temperature ^{FCC } ^{structure } exhibit order-disorder low temperature phase transitions. The model system ^{of such } ^{a } behaviour is AuCu [1, 2]. ^{This } system ^{has low} temperature L12 ^{and } Llo structures in the range of 25 % and 50 % atomic concentrations, respectively.

### The observed phase diagram ^{can } ^{here } ^{be } roughly ^{described } by ^{an } Ising nearest-neighbour

### Hamiltonian [3].

### Since this Hamiltonian leads to strong degeneracies of the ordered states at T

^{= }

### 0, ^{it is } ^{an}

### excellent tool to compare the various approximations ^{used } ^{to } calculate the entropy ^{of the} phases ^{at } ^{T #: 0.}

### Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500190298300

### 2984

### The phase diagram obtained from (1) has been calculated by ^{the } Bragg-Williams ^{mean } ^{field} approximation [4], by the cluster variational method [5, 6], by ^{low and } high temperature expansions [7] ^{and } by the Monte Carlo method [3, ^{8, 9, } 10].

### Close to the 25 % and 50 % concentrations, ^{the } degeneracy appears among the various

### antiphase structures built from L12 ^{and } Llo with any ^{set } of (1/2 1/2 0) antiphase ^{vectors } ^{and}

### with the antiphase planes perpendicular ^{to } ^{the } (0 0 1) ^{direction } (see Fig. 1). Some of these

### antiphase structures are observed in the CuAu system ^{close } ^{to } ^{the } ordering temperature [2].

### The aim of this paper is ^{to } show how the Monte Carlo method can provide precise ^{values of}

### the free energy of these various structures and to compare their relative stability. ^{The}

### notation used in the study of ANNNI model (see ^{Ref. } [11]) is also used here : 00 >

### corresponds ^{to } ^{no } antiphase, (i j... ) corresponds ^{to } ^{a } periodic ^{structure } ^{with } ^{two } antiphases separated by i periods of the FCC lattice and the next two separated by j periods...

### Fig. 1.

^{- }

^{Sketch } ^{of } ^{some } structures studied.

### 1. Description of the calculations.

### The general method has been described in [12], following ^{a } previous work of Vanneau (see

### Ref. [13]).

### 1.1 M.C. SIMULATIONS.

^{- }

### All calculations reported ^{here } ^{are } ^{carried } out at constant atomic concentrations (Canonical Ensemble) ^{on a } lattice of N

^{= }

### 2 048 atoms. This lattice is initially

### in one of the completely ^{ordered } structures studied (« ground states ») L12 (25 %) ^{or} Llo (50 % ) ; ^{and the } (11), (22), (44 ) ^{antiphase } structures corresponding ^{to } ^{these } ^{two}

### concentrations. These are the only structures which can be studied here (L

^{= }

^{8 } ^{with}

### 2 048 FCC atoms).

### The Metropolis algorithm [14] ^{is used } ^{to }

^{« }

### equilibrate

^{» }

^{the } system ^{at } ^{a } given temperature ^{T} during ^{7 } 000 Monte Carlo steps (MCS).

### The energy E of the system ^{is then } regularly sampled (E

^{= }

^{0 in the } completely ^{ordered} states). Samples ^{are } carried among ^{at } least 100 000 MCS (up ^{to } ^{1000 } ^{000 } MCS).

### The 28 first short range order parameters ^{are } ^{also } calculated, ^{in order } ^{to } verify ^{the}

### metastability of the initial structure. It is observed that the systems studied remain metastable if Ts 0.95 To (assuming kTo/J ^{is 1.81 } ^{at } ^{25 } ^{% } ^{and 1.73 } ^{at } ^{50 } %). ^{At } higher temperatures,

### irregular antiphases appear, apparently destroying ^{the } long range order, though ^{the } energies

### remain in the same range.

### The number of states of each energy is counted and the result is the normalized distribution

### f T(E) ^{for each } structure. This distribution can be written :

### where g (E) is the total number of states of energy E of the ^{structure } and Z (T) ^{is the } partition

### function ((3

^{= }

### 1 /kB T) ; g (E) is. the density ^{of } ^{states } ^{of the } ^{structure } ^{studied.}

### From formula (2), g(E) can be obtained and all thermodynamic ^{functions } ^{can } ^{be } deduced,

### if Z(T) is known. Figure 2 shows the distributions fT(E) computed ^{in the } L12 structure at three temperatures (1.4 J, ^{1.65 } J, 1.72 J).

### Fig. ^{2. }

^{- }

### Distributions f T(E) ^{in the } L12 (25 %) structures at kT/J ^{= 1.4, } 1.6 and 1.72.

### 1.2 CALCULATION OF Z(T).

^{- }

### Z(T) ^{can } be calculated from formula (2) if any ^{exact } value of

### g (E) is known :

### In practice, gex(E) ^{can } only be calculated at low E and f T (E ) ^{has } significant ^{values } ^{at } ^{these} energies ^{in the } ^{case } of the lowest temperatures ^{studied } (see Fig. 2).

### At these temperatures, ^{the } f T(E) obtained with the various antiphase structures do not differ. For this reason, Z(T) is here assumed to be the same.

### On the other hand, gex(E) ^{has } ^{a } polynomial expression ^{in N. The } higher ^{order } ^{terms } ^{of the}

### 12 first polynomials ^{are } exactly calculated and their next terms are estimated from Monte Carlo calculations with small systems (i.e. ^{N } ^{2 } 048). The latter terms introduced only ^{a } ^{few} percent correction on geX (E). In every ^{case } g (0 ) ^{= 1.}

### A least squares fit method has been used ^{to } determine Z in the finite system studied from

### 2986

### formula (3) :

### The values of Z ( T) ^{at } higher temperatures ^{are } deduced from the

^{« }

### overlap ^{method }

^{». }

^{If } ^{two}

### distributions fT(E) ^{obtained } ^{at } ^{T and T’ } overlap :

### In the case studied (see Fig. 2) ^{three } temperatures provide ^{an } ^{excellent } overlap. ^{Table 1}

### shows the values of 11N ^{Log } (Z(T)/Z(T’)) ^{obtained } by ^{the least } squares fit method.

### Table 1.

^{- }

### 1 IN Log Z (T)IZ (T’) ^{in } ^{the } different structures studied.

### 1.3 DENSITY OF STATES.

^{- }

### The density ^{of } ^{states } g (E) ^{of each } ^{structure } is deduced from formula (2). ^{In } practice, the function Ig (e) ^{is } tabulated :

### with e

^{= }

### E/NJ.

### Figure 3 shows the results obtained in the L12 (25 %) structure. The relative errors can be close to 3 x 10- 4 and only 150 different energy ^{states } ^{are } explored in the ordered systems ^{with} 2 048 atoms. It is observed that the values of Ig (s) ^{are } ^{about the } ^{same } for the various

### antiphases ^{if E } ^{0.08. At } increasing E, increasing differences appear between the

### L12 ^{and } Llo phases (higher Ig ( E )) ^{and the } antiphase ^{ones } (lower Ig (e». These differences remain nevertheless small : 25 (2) ^{x } ^{10- 5 } between the L12 and (11) (D022) structures if

E =

### 0.2 at 25 %.

### 1.4 LINEAR EXTRAPOLATION OF Ig ( E ). - In all ordered structures (an exemple ^{is } given ^{in}

### Fig. 3), Ig (s) ^{exhibits } ^{a } linear behaviour at e

> ### 0.21 :

### Fig. ^{3. }

^{- }

^{Plot of } Ig (e)

^{= }

^{Ln } (g (e ))/ N ^{for the } ^{L12 } (25 %) ^{structure.}

### The coefficients

^{« }

### and y have been calculated by ^{a } least squares fit from e

^{= }

### 0.21 to

8 ^{= }

### 0.29. The results are given in table II.

### Table II.

^{- }

### Linear extrapolations of Ig (e) ^{used } ^{in } formula (6).

### In this table, ^{the }

^{« }

^{HT » } (High Temperature) ^{values } correspond ^{to } ^{a } precise re-calculation of the free energy in the disordered system. ^{In } ^{this } calculation, ^{the } ^{same } overlapping ^{method}

### has been used as in [12], ^{but } Z(T) has been corrected from finite size effects :

### The correction term is two orders of magnitude higher ^{than } ^{the } ^{errors } of the calculation. The

### same linear behaviour as (6) ^{is } ^{observed } ^{on } ^{the } ^{curve } Ig ( E ) ^{at }

^{« }

### HT » if 0.5

^{E }

### 0.58.

### In all _{cases, } linear extrapolations ^{of } Ig (e) ^{with } (6) have been used from e

^{= }

### 0.24 to

E ^{= }

### 0.58. The validity ^{of these } extrapolations is further discussed.

### 2. Results.

### From Ig (e), ^{all } thermodynamic ^{functions } ^{are } ^{deduced } precisely ^{at } ^{all } temperatures T 0.99 To.

### Figures ^{4 } and 5 show the results obtained at 25 % and 50 % respectively. ^{The free } energies

### F/N ^{are } always smaller in thé (oo) ^{states. } The differencies observed are low (see ^{Tab. } 1).

### 2988

### Fig. ^{4. }

^{- }

### Free energy, entropy and energy ^{at } 25 % composition.

### Fig. ^{5. }

^{- }

### Free energy, entropy and energy ^{at } 50 % composition.

### Only the differences between the oo > and 11 > structures are large enough ^{to } be visible in the figures given ^{and } only ^{close } ^{to } To. On the other hand, e (T) ^{and } S(T)/kN ^{are more}

### contrasted. In these figures

^{« }

### HT » results are also given.

### 3. Antiphase ^{free } energy.

### The difference in free energy per ^{atom } 03B4F(i) ^{between } ^{an } ^{antiphase } structure i i >

### (i = 1, 2, 4 ) ^{and } the oo > phase ^{is here } always positive ^{and it } ^{can } ^{be } expressed ^{as a } ^{surface}

### Fig. 6. Fig. 7.

### Fig. ^{6. }

^{- }

### Antiphase free energy f S (i , T) ^{at } ^{25 } ^{%.}

### Fig. ^{7. }

^{- }

### Antiphase free energy fs(1 , T ) ^{at } ^{50 %.}

### free _{energy :}

### if the unit surface is a2 (a ^{is the } ^{lattice } parameter ^{of the }

^{« }

^{HT » } ^{FCC } lattice). Figures ^{6 and 7}

### show the variations of fs(1 ) ^{versus } T and i for the two concentrations studied.

### 4. Discussion.

### CVM [6] ^{and low } temperature expansions [15] have shown that the 00) structures are more

### stable than the (11) structures. This result compares well with, ^{for } instance, ^{CVM } ^{result} given ^{in } [6] : 03B4F (1 )/J

^{= }

^{8.78 } ^{x } ^{10-4 } ^{at } kB ^{T } = 1.8 and C

^{= }

### 25 %. The present ^{result is}

### 6F(1)/7= 1.1 ^{x 10- 3.}

### It appears here that the 00) structures are always ^{the } ^{most } ^{stable } ^{ones. } ^{In the } ^{case } ^{of the} Llo structure, fs(1) ~ fs(2). ^{In } ^{the } ^{case } ^{of } L12, fs(2) ^{is } significantly higher ^{than} f s (1 ). ^{As } the lattice used in our Monte Carlo study ^{is } relatively small, ^{the } given ^{values of} fs(4) ^{are } ^{not } very reliable.

### Now, ^{let } ^{us } discuss the linear behaviour of Ig (s) observed in ordered systems ^{if} 0.2

^{E }

### 0.29 and in disordered systems ^{if } ^{0.5 }

^{s }

^{0.58 } (see ^{also } [16]).

### The slopes y of Ig ( E ) ^{for the } ^{ordered } structures (Tab. II) ^{are } higher ^{in } the 00) structures.

### In the latter case, they ^{are } very close (although lower) ^{to } ^{the }

^{« }

^{HT }

^{» }

### (disordered) ^{values of } ^{y.}

### On the other hand, ^{the } corresponding values of

^{a }

### are also the closest between the

### 00) ^{and the }

^{« }

^{HT }

^{» }

### values. These results suggest that, though ^{the } ^{0.29 }

^{e }

^{0.5 } region ^{is}

### not explored ^{here, } there exists a continuity ^{of } Ig (s) between the fluctuations of

### (oo) ^{ordered } ^{states } and the « HT » fluctuations. The shape of this continuous function

### lgo (e) ^{should } then be linear in the whole 0.21

^{E }

### 0.58 _{range. } This brings ^{a } justification ^{of}

### the linear extrapolations ^{uses } ^{in } ^{our } calculations. It is suggested ^{in } [16] that this behaviour is related to the first order character of the transition studied here.

### If this function lgo ( E ) is linear from _{El = } 0.21 to E2 0.58, ^{it } ^{can } ^{be } easily ^{shown that}

### table II gives ^{estimates } ^{of :}

### Log (Z (To ) )/N

^{= }

^{a, } J/kB To

^{= }

### y and of the energy jump ^{at } ^{the } transition: à e = 03B52 - E1.

### From table II, kB To/J =1. 81 ^{at } 25 % and 1.73 at 50 %.

### In a study ^{with } only ^{2 } 048 atoms, ^{the } hydrodynamic slowing ^{down } ^{at } ^{the } transition is not

### strong : ^{no } hysteresis is observed here. Moreover, it is observed that if the disordered states

### are sampled ^{at } To ^{T } ^{1.02 } To, ^{the } ^{curve } f T (E ) ^{has } ^{a }

^{« }

^{double } hump » shape (sometimes

### called a « camel’s back »). Here, ^{it } happens ^{that } J/kB T is intermediate between yyçr (or y (oo) ^{and } yantiphases (see ^{Tab. } II). ^{In such } ^{a } ^{case, } ^{if } 03B51

^{ E }

### E2, the Monte Carlo

### sampling explores ^{two } types ^{of } ^{states : } (i) disordered states with energy probabilities proportional ^{to } exp ((y HT - 03B2 ) E) (increasing exponential) ^{and } (ii) ^{ordered } ^{states } ^{with}

### numerous antiphases, ^{whose } probability distribution is close to a decreasing exponential. ^{The}

### latter states have a strong probability ^{of } occurring ^{in } ^{a } ^{finite } system. ^{For } instance, ^{it } ^{can } ^{be}

### shown in our case (L

^{= }