Free energy of low temperature antiphase structures in a FCC nearest-neighbour antiferromagnetic Ising model by the Monte Carlo method

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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

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

hamiltonien

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

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}

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

fonction de la température ^et de la distance entre

antiphases. ^A partir des résultats obtenus,

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

Abstract.

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

modelled by

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

antiphases. The surface free energy of the antiphases is calculated and its variations

versus

T and

the distance between antiphases

studied. From the results obtained,

discussion is carried out of the various

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

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).

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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

0.29 and in disordered systems ^{if 0.5}

^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

are also the closest between the

00) ^{and the}

^HT

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

^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

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

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

8 ) ^{that the} ( 11 ) structures, having ^a higher multiplicity, ^{are more} probable ^at every T To ^than the 00) ^structure.

The occurrence of the antiphase structures, if the disordered system ^is slowly cooled, ^leads

to overestimate of the ordering temperatures (as ⁱⁿ [8]). ^{The most stable} phase ^{cannot be}

obtained by ^{this method} and the free energy of the ordered structures cannot be calculated.

The effects described above should decrease on increasing ^L. Unfortunately, ^{due to the} hydrodynamic showing down, ^the exploration ^of Ig (E) ^between el and E2 becomes then

extremely ^difficult.

2990

Acknowledgments

The author wants to thank A. Finel, ^F. Ducastelle, ^C. Bichara, ^{J. P.} Gaspard, G. Martin and F. Bley ^for having stimulated this work.

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