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)



Les alliages CFC à mise


ordre (25 et 50 %) sont modélisés par



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

ceux sans

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.



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


T and


the distance between antiphases


studied. From the results obtained,


discussion is carried out of the various


of the Monte Carlo method.


Physics Abstracts







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



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]).



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




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.



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



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.



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

with e



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



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).



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




(disordered) values of y.

On the other hand, the corresponding values of


are also the closest between the

00) and the




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.




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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